Variational Analysis of the $J_1$–$J_2$–$J_3$ Model: A Non-linear Lattice Version of the Aviles–Giga Functional

Abstract

We study the variational limit of the frustrated $J_1$–$J_2$–$J_3$ spin model on the square lattice in the vicinity of the ferromagnet/helimagnet transition point as the lattice spacing vanishes. We carry out the $\Gamma$-convergence analysis of proper scalings of the energy and we characterize the optimal cost of a chirality transition in BV proving that the system is asymptotically driven by a discrete version of a non-linear perturbation of the Aviles–Giga energy functional.

1 Introduction

Low-energy states of two-dimensional magnetic compounds feature a large variety of complex magnetic patterns. The emergence of some of these structures is usually the result of a number of competing interactions whose relative weight may drastically change with the length scale. From the physical point of view, the resulting unconventional magnetic order often corresponds to a rich phase diagram. The experimental community has recently made great progresses in unveiling critical properties of such phase diagrams. In addition, in the statistical mechanics community there has been a quest for elementary lattice spin models that would reproduce some of the most surprising geometric patterns of low-energy states introducing a minimal number of parameters in the model (see [27] and the references therein for a recent overview on this topic). One of the key features of such energetic models is the frustration mechanism, that is, roughly speaking, the presence of conflicting interatomic forces that prevent the energy of every pair of interacting spins to be simultaneously minimized. In the recent years, several examples of frustrated spin models have been investigated from a variational perspective, cf. [1, 9, 10, 15, 17, 21, 23, 29,30,31]. As these examples show, the presence of frustration in a lattice spin system depends on both the topological properties of the lattice and the symmetry properties of the interaction potentials.

In this paper we are going to investigate a model in which frustration originates from the competition of ferromagnetic (F) and antiferromagnetic (AF) interactions. This model is known as the $J_{1}$–$J_{2}$–$J_{3}$ F-AF classical spin model on the square lattice (see, e.g., [49]). To each configuration of two-dimensional unitary spins on the square lattice, namely $u :\mathbb {Z}^2 \rightarrow \mathbb {S}^1$, we associate the energy

$$\begin{aligned} E(u) = - J_1 \sum _{|\sigma - \sigma '|=1} u^{\sigma } \cdot u^{\sigma '} + J_2 \sum _{|\sigma - \sigma ''|=\sqrt{2}} u^{\sigma } \cdot u^{\sigma ''} + J_3 \sum _{|\sigma - \sigma '''|=2} u^{\sigma } \cdot u^{\sigma '''}, \end{aligned}$$

where $J_1$, $J_2$, and $J_3$ are positive constants (the interaction parameters of the model) and for every lattice point $\sigma \in \mathbb {Z}^2$ we let $u^\sigma$ denote the value of the spin variable u at $\sigma$. The energy consists of the sum of three terms. The first is ferromagnetic as it favors aligned nearest-neighboring spins, whereas the second and the third one are antiferromagnetic as they favor antipodal second-neighboring and third-neighboring spins, respectively.

In the case where $J_2 = J_3 = 0$, the energy above describes the so-called XY model, a ferromagnetic model which can be considered a lattice version of the Ginzburg-Landau model for type II superconductors. The latter is an energy functional which has drawn the attention of the mathematical community since several decades (see, e.g., [12, 52] and the references therein) and which shares with the XY functional many similarities as pointed out in [3]. The variational analysis of the XY model has been carried out in [2] also in connection to the theory of dislocations [4, 48]. We also mention the more recent results in [16] on a variant of the XY model on a non-flat lattice and the results in [18,19,20] regarding its connections with the N-clock model.

In the case in which $J_2 = 0$ and $J_3 > 0$, E becomes the energy of the $J_1$–$J_3$ model considered in [17]. In that paper, it has been shown that the ferromagnetic and the antiferromagnetic terms in E compete and give rise to ground states in the form of helices of possibly different chiralities (for recent experimental evidences on helical ground states of the $J_1$–$J_3$ and of the $J_1$–$J_2$–$J_3$ models see, e.g., [53, 55]). Referring the energy E to that of such helimagnetic ground states, one can then investigate the energetic behavior of low-energy spin configurations in a bounded domain as the lattice spacing vanishes. In terms of $\Gamma$-convergence, one can prove the existence of a specific energy scaling at which chirality transitions take place and describe the energetic behavior of the system in terms of an effective macroscopic energy which gives the cost of such chirality transitions. The goal of this paper is to follow a similar approach in the complete $J_1$–$J_2$–$J_3$ model. We explain this approach below in more details.

To study the asymptotic variational limit of the energy E as the number of particles diverges, we consider the sequence of energies $E_n$ obtained as follows: We fix a bounded open set $\Omega \subset \mathbb {R}^2$ and we scale the lattice spacing by a small parameter $\lambda _n> 0$. Given $u :\lambda _n\mathbb {Z}^2 \cap \Omega \rightarrow \mathbb {S}^1$, writing $\sigma \in \mathbb {Z}^2$ in components as (i, j), and letting $u^{i,j}$ denote the value of u at $(\lambda _ni, \lambda _nj)$, the energy per particle in $\Omega$ reduces to the sequence of energies

$$\begin{aligned} \begin{aligned} E_n(u)&:= - \alpha \lambda _n^2 \sum _{(i,j)} \Big ( u^{i,j} \cdot u^{i+1,j} + u^{i,j} \cdot u^{i,j+1} \Big ) \\&\quad + \beta \lambda _n^2 \sum _{(i,j)} \Big ( u^{i,j} \cdot u^{i+1,j+1} + u^{i,j} \cdot u^{i-1,j+1} \Big ) \\&\quad + \lambda _n^2 \sum _{(i,j)} \Big ( u^{i,j} \cdot u^{i+2,j} + u^{i,j} \cdot u^{i,j+2} \Big ), \end{aligned} \end{aligned}$$

(1.1)

where $\alpha =J_1/J_3$, $\beta = J_2/J_3$, and the sums are taken over all those $(i,j) \in \mathbb {Z}^2$ for which all evaluations of u above are defined.

We are interested in the case where the parameters $\alpha$ and $\beta$ depend on the lattice spacing $\lambda _n$, hence we write $\alpha = \alpha _n$ and $\beta = \beta _n$. We focus on the range $0 \leqq \beta _n\leqq 2$ and we note that, depending on the parameter $\alpha _n$, the ground states of the system are either ferromagnetic or helimagnetic as depicted in the phase diagram reported in Fig. 1 (cf. also [49, Figure 2]). To explain the emergence of the different types of ground states, it is convenient to rewrite the energy $E_n(u)$ (up to an additive constant and neglecting error terms at the boundary of $\Omega$) as

Fig. 1

A schematic representation of the case studied in this paper. For $\beta \in [0,2]$, the line $\beta = \frac{\alpha - 4}{2}$ separates the cases where the ground states are helimagnetic and ferromagnetic. We are interested in helimagnet/ferromagnet transitions, i.e., in the case where the values $(\alpha ,\beta )$ approach the aforementioned line. The boundary case $\beta \equiv 0$ corresponds to the so-called $J_1$–$J_3$ model, whose variational analysis at the helimagnet/ferromagnet transition point $\alpha \rightarrow 4$ has been carried out in [17]. In this paper, we examine in detail the opposite boundary case $\beta \equiv 2$ when $\alpha$ approaches the value 8. The main features of the in-between cases $\beta \in (0,2)$ can be obtained by combining the behaviors in the two extreme cases, see Remark 4.6

$$\begin{aligned} \begin{aligned} F_n(u)&:= \frac{\beta _n}{4} \lambda _n^2 \sum _{(i,j)} \Big | u^{i+1,j} - \frac{\alpha _n}{\beta _n+2} u^{i,j} + u^{i-1,j} + u^{i,j+1} - \frac{\alpha _n}{\beta _n+2} u^{i,j} + u^{i,j-1} \Big |^2 \\&\quad + \frac{2-\beta _n}{4} \lambda _n^2 \sum _{(i,j)} \Big | u^{i+1,j} - \frac{\alpha _n}{\beta _n+2} u^{i,j} + u^{i-1,j} \Big |^2 \\&\quad + \Big | u^{i,j+1} - \frac{\alpha _n}{\beta _n+2} u^{i,j} + u^{i,j-1} \Big |^2; \end{aligned} \end{aligned}$$

(1.2)

we refer to Section 2.5 for the details. If the ferromagnetic nearest-neighbor interaction parameter $\alpha _n$ is large enough, one expects the ferromagnetic order to dominate, leading to ground states made of parallel spins $u \equiv \mathrm {const.} \in \mathbb {S}^1$. The range of all $\alpha _n$ leading to this behavior is characterized by the inequality $\frac{\alpha _n}{\beta _n+2} \geqq 2$, which can be explained by the following simple heuristic argument. One starts by observing that, for $\frac{\alpha _n}{\beta _n+2} = 2$, ferromagnetic states are the only spin configurations which make $F_n$ zero. As a consequence, since larger values of $\frac{\alpha _n}{\beta _n+2}$ increase the weight of the ferromagnetic interactions versus the antiferromagnetic interactions even more, ferromagnetic ground states should appear also for $\frac{\alpha _n}{\beta _n+2}>2$. A rigorous proof of this argument is based on a simple comparison argument already used in the one-dimensional case investigated in [21] and that can be repeated in the present case verbatim. If instead $\frac{\alpha _n}{\beta _n+2} < 2$, the ground states have a different geometry. If $\beta _n< 2$, they are completely characterized by the requirement that all the squares in (1.2) are zero. This can be achieved only by choosing a helical spin field $u :\lambda _n\mathbb {Z}^2 \cap \Omega \rightarrow \mathbb {S}^1$ such that

$$\begin{aligned} u^{i,j} = \big (\cos (\theta _0 + i \theta ^{\mathrm {hor}}+ j \theta ^{\mathrm {ver}}) \ , \ \sin (\theta _0 + i \theta ^{\mathrm {hor}}+ j \theta ^{\mathrm {ver}}) \big ), \end{aligned}$$

(1.3)

where $\theta ^{\mathrm {hor}}, \theta ^{\mathrm {ver}}\in \big \{ \pm \arccos \big (\frac{\alpha _n}{2(\beta _n+ 2)} \big ) \big \}$ and $\theta _0 \in [0, 2 \pi )$. Indeed, for such a spin field we have that

$$\begin{aligned} u^{i+1,j} + u^{i-1,j} = u^{i,j+1} + u^{i,j-1} = \frac{\alpha _n}{\beta _n+2} u^{i,j}. \end{aligned}$$

The four possible families of ground states obtained by choosing the signs of $\theta ^{\mathrm {hor}}$ and $\theta ^{\mathrm {ver}}$ correspond to left-handed or right-handed helices directed along the lattice rows or columns, respectively. A concise description of this discrete ground state degeneracy is made possible by introducing the notion of chirality vector $\chi$. Roughly speaking, $\chi$ represents the direction along which the helical configuration is rotating most and is given by

$$\begin{aligned} \chi \simeq \frac{1}{ \sqrt{2} \arccos \big (\tfrac{\alpha _n}{2(\beta _n+ 2)} \big ) } (\theta ^{\mathrm {hor}}, \theta ^{\mathrm {ver}}), \end{aligned}$$

(1.4)

i.e., by normalizing the vector $(\theta ^{\mathrm {hor}}, \theta ^{\mathrm {ver}})$ of the angles between horizontally and vertically adjacent spins.¹ According to this definition, the four families of ground states in the regime $\frac{\alpha _n}{\beta _n+2} < 2$, $\beta _n< 2$, correspond to $\chi$ taking one of the four values

$$\begin{aligned} \frac{1}{\sqrt{2}} (+1,+1), \ \frac{1}{\sqrt{2}} (+1,-1)\, , \ \frac{1}{\sqrt{2}} (-1,+1), \ \frac{1}{\sqrt{2}} (-1,-1) \, \end{aligned}$$

(1.5)

(see, e.g., the second picture in Fig. 2 for an illustration of the value $\frac{1}{\sqrt{2}} (-1,+1)$). When $\beta _n= 2$ and $\frac{\alpha _n}{\beta _n+2} < 2$, ground states only need to satisfy the weaker condition

$$\begin{aligned} u^{i+1,j} + u^{i-1,j} + u^{i,j+1} + u^{i,j-1} = \frac{2 \alpha _n}{\beta _n+2} u^{i,j} = \frac{\alpha _n}{2} u^{i,j}. \end{aligned}$$

This can be achieved by helical fields as in (1.3) with $\theta ^{\mathrm {hor}}, \theta ^{\mathrm {ver}}$ satisfying the relation $\cos (\theta ^{\mathrm {hor}}) + \cos (\theta ^{\mathrm {ver}}) = \frac{\alpha _n}{4}$. The latter condition is equivalent to requiring the chirality vector to have unitary length, namely $\chi \in \mathbb {S}^1$. Fig. 2 shows the helical ground state u corresponding to different choices of $\chi \in \mathbb {S}^1$.

In this paper we investigate the chirality properties of spin fields with low $J_1$–$J_2$–$J_3$ energy for a choice of parameters corresponding to spin configurations close to the helimagnet-ferromagnet transition point. This is equivalent to assuming that $0 \leqq \beta _n\leqq 2$ and that $2(\beta _n+ 2) - \alpha _n\searrow 0$. Within this range of parameters, the asymptotic behavior of (an appropriate scaling of) $F_n$ is established by rewriting the energy in terms of a microscopic notion of chirality that we associate to any admissible spin configuration. Such a chirality (still denoted by) $\chi$ will then be a discrete vector field defined on $\lambda _n\mathbb {Z}^2 \cap \Omega$, the order parameter of the system.

In the case $\beta _n\equiv 0$, this program has already been carried out in [17]. In that paper, it has been proved that transitions in the chirality parameter $\chi$ cost an energy of order $(4 - \alpha _n)^{3/2} \lambda _n$. Moreover, expressed in terms of $\chi = (\chi _1 , \chi _2)$, the accordingly scaled energies $((4-\alpha _n)^{-3/2} \lambda _n^{-1}) F_n$ behave like a functional of the form

$$\begin{aligned} \frac{1}{2} \int \frac{1}{\varepsilon _n} \Big ( \big |\tfrac{1}{2} - |\chi _1|^2 \big |^2 + \big |\tfrac{1}{2} - |\chi _2|^2\big |^2 \Big ) + \varepsilon _n\big ( |\partial _1 \chi _1|^2 + |\partial _2 \chi _2|^2 \big ) \, {\mathrm {d}{x}}, \end{aligned}$$

where $\varepsilon _n\simeq (4-\alpha _n)^{-\frac{1}{2}}\lambda _n\rightarrow 0$. In addition, the crucial observation that $\chi$ is forced to be approximately a curl-free vector field, say $\chi \simeq \nabla \varphi$, has made possible to recognize the functional above as a Modica-Mortola type functional written in the gradient variable $\nabla \varphi$. This functional features a four-well potential, whose zeros correspond to the four possible chiralities of the ground states mentioned in (1.5). Exploiting these observations it has been proved that the $\Gamma$-limit of $((4-\alpha _n)^{-3/2} \lambda _n^{-1}) F_n$ is finite on $BV\big (\Omega ; \big \{ \pm \frac{1}{\sqrt{2}}\big \}^2\big )$ chiralities with vanishing curl and takes the form of an interfacial energy between regions with different constant chiralities.

It can be observed that the full $J_1$–$J_2$–$J_3$ model shares similarities with the $J_1$–$J_3$ model mentioned above, if $\sup _n \beta _n< 2$, see Remark 4.6 below. (This is related to the fact that, as in the $J_1$–$J_3$ model, ground states of the $J_1$–$J_2$–$J_3$ energy can only have one of the four possible chiralities in (1.5) for all $\beta _n< 2$.) If, instead, $\beta _n\rightarrow 2$, the behavior of the $J_1$–$J_2$–$J_3$ system can be substantially different. To single out the new features of the model, in this paper we consider the extreme case $\beta _n\equiv 2$. In Remark 4.6 we explain how to obtain a satisfactory description of the model in more general cases by combining the analysis of the case $\beta _n\equiv 0$ examined in [17] with the results in the case $\beta _n\equiv 2$. With this particular choice of $\beta _n$, the helimagnet-ferromagnet transition point we are interested in corresponds to $\alpha _n\nearrow 8$.

Fig. 2

Three examples of ground states of the $J_1$–$J_2$–$J_3$ model. The three ground states are distinguished by different chirality vectors that set the speed of rotation of the spin in the horizontal and vertical direction. The chirality vector can be any direction in $\mathbb {S}^1$

Our analysis of the case $\beta _n\equiv 2$ is made possible by the key observation that, written in terms of $\chi$, suitable rescalings of $F_n$ resemble a discrete version of the Aviles–Giga functional. In the following we present a heuristic computation which motivates such an analogy, referring to Section 2.6 for a more rigorous derivation. Let us introduce the small parameter $\delta _n:= 4 - \frac{\alpha _n}{2}$ which we will also use throughout the paper. Roughly speaking, an angular lifting $\psi$ such that $u = (\cos \psi , \sin \psi )$ is related to the angles $\theta ^{\mathrm {hor}}$ and $\theta ^{\mathrm {ver}}$ between horizontally and vertically neighboring spins via $(\theta ^{\mathrm {hor}}, \theta ^{\mathrm {ver}}) \simeq \lambda _n\nabla \psi$. According to that, in view of (1.4) (for $\beta _n= 2$), we can write

$$\begin{aligned} \chi \simeq \frac{\lambda _n}{\sqrt{2} \arccos \big ( 1 - \tfrac{\delta _n}{4} \big )} \nabla \psi \simeq \frac{\lambda _n}{\sqrt{2} \sqrt{\tfrac{\delta _n}{2}}} \nabla \psi = \nabla \varphi , \end{aligned}$$

where we have set $\varphi := \frac{\lambda _n}{\sqrt{\delta _n}} \psi$. To rewrite $F_n$ in terms of $\chi$, for $\lambda _n$ small enough, we may write $(u^{i+1,j} - 2 u^{i,j} + u^{i-1,j})/\lambda _n^2 \simeq \partial _{11} u$ and $(u^{i,j+1} - 2 u^{i,j} + u^{i,j-1})/\lambda _n^2 \simeq \partial _{22} u$. Therefore,

$$\begin{aligned} F_n(u)= & {} \frac{1}{2} \lambda _n^2 \sum _{(i,j)} \Big | u^{i+1,j} + u^{i-1,j} + u^{i,j+1} + u^{i,j-1} - \frac{\alpha _n}{2} u^{i,j} \Big |^2 \\= & {} \frac{1}{2} \lambda _n^2 \sum _{(i,j)} \Big | \delta _nu^{i,j} + \lambda _n^2 \frac{u^{i+1,j} - 2u^{i,j} + u^{i-1,j}}{\lambda _n^2} \\&+ \lambda _n^2 \frac{u^{i,j+1} - 2 u^{i,j} + u^{i,j-1}}{\lambda _n^2} \Big |^2 \\\simeq & {} \frac{1}{2} \int \delta _n^2 + 2 \lambda _n^2 \delta _nu \cdot (\partial _{11} u + \partial _{22} u) +\lambda _n^4 | \partial _{11} u + \partial _{22} u |^2 \, {\mathrm {d}{x}} \\= & {} \frac{1}{2} \int \delta _n^2 + 2 \lambda _n^2 \delta _nu \cdot \Delta u + \lambda _n^4 | \Delta u |^2 \, {\mathrm {d}{x}}. \end{aligned}$$

We observe that $u \cdot \Delta u = - |\nabla \psi |^2$ and $|\Delta u|^2 = |\nabla \psi |^4 + |\Delta \psi |^2$. As a consequence, the above integral reads

$$\begin{aligned}&\frac{1}{2} \int \delta _n^2 - 2 \lambda _n^2 \delta _n|\nabla \psi |^2 + \lambda _n^4 |\nabla \psi |^4 + \lambda _n^4 |\Delta \psi |^2 \, {\mathrm {d}{x}} \\&\quad = \frac{1}{2} \int \big | \delta _n- \lambda _n^2 |\nabla \psi |^2 \big |^2 + \lambda _n^4 |\Delta \psi |^2 \, {\mathrm {d}{x}}. \end{aligned}$$

Thus,

$$\begin{aligned} F_n(u)\simeq & {} \frac{1}{2} \int \delta _n^2 \big | 1 - |\nabla \varphi |^2 \big |^2 + \lambda _n^2 \delta _n|\Delta \varphi |^2 \, {\mathrm {d}{x}}\\= & {} \delta _n^{3/2} \lambda _n\frac{1}{2} \int \frac{1}{\varepsilon _n} \big | 1 - |\nabla \varphi |^2 \big |^2 + \varepsilon _n|\Delta \varphi |^2 \, {\mathrm {d}{x}}, \end{aligned}$$

where we have set $\varepsilon _n= \frac{\lambda _n}{\sqrt{\delta _n}}$. To make these computations rigorous, in Section 2.6 we introduce the functionals $H_n(\chi , \Omega ) \simeq \frac{1}{\delta _n^{3/2} \lambda _n} F_n(u)$. These resemble a discretization of the functionals

$$\begin{aligned} AG^\Delta _{\varepsilon _n}(\varphi , \Omega ) := \frac{1}{2} \int _{\Omega } \frac{1}{\varepsilon _n} \big | 1 - |\nabla \varphi |^2 \big |^2 + \varepsilon _n|\Delta \varphi |^2 \, {\mathrm {d}{x}}, \end{aligned}$$

(1.6)

where $\chi \simeq \nabla \varphi$. The latter are variants of the classical Aviles–Giga functionals

$$\begin{aligned} AG_{\varepsilon _n}(\varphi , \Omega ) := \frac{1}{2} \int _{\Omega } \frac{1}{\varepsilon _n} \big | 1 - |\nabla \varphi |^2 \big |^2 + \varepsilon _n|\nabla ^2 \varphi |^2 \, {\mathrm {d}{x}} \end{aligned}$$

(1.7)

and share with them most of their properties related to their $\Gamma$-convergence as $\varepsilon _n\rightarrow 0$. We will study the asymptotic properties of the functionals $H_n$ for $\lambda _n\ll \sqrt{\delta _n}$, the regime which corresponds to $\varepsilon _n\rightarrow 0$.

The sequence of Aviles–Giga functionals has been introduced by Aviles and Giga [7] and Gioia and Ortiz [45] to study smectic liquid crystals and blistering in thin films. Although similar in form to the sequence of Ginzburg-Landau functionals, its asymptotic behavior as $\varepsilon \rightarrow 0$ is completely different due to the curl-free constraint on the vector field $\nabla \varphi$. In [7] it has been conjectured that the $\Gamma$-limit as $\varepsilon \rightarrow 0$ of $AG_{\varepsilon }$ is a functional finite on functions $\varphi \in W^{1,\infty }(\Omega )$ solving the eikonal equation

$$\begin{aligned} |\nabla \varphi | = 1 \text { a.e. in } \Omega \end{aligned}$$

(1.8)

and charges jumps of the gradient field $\nabla \varphi$. The analysis of one-dimensional transition profiles suggests that the $\Gamma$-limit behaves as the defect energy

$$\begin{aligned} \frac{1}{6} \int _{J_{\nabla \varphi }} |[\nabla \varphi ]|^3 \, {\mathrm {d}{\mathcal {H}}}^{1} \, , \end{aligned}$$

(1.9)

where $J_{\nabla \varphi }$ is the jump set of $\nabla \varphi$, $[\nabla \varphi ](x)$ is the jump of $\nabla \varphi$ at $x \in J_{\nabla \varphi }$, and $\mathcal {H}^1$ is the one-dimensional Hausdorff measure.

If one assumes that $\varphi$ belongs to the set of functions solving (1.8) and such that $\nabla \varphi \in BV(\Omega )$, then it has been proved (cf. [5, 8, 22, 34, 46]) that $AG_\varepsilon$ $\Gamma$-converge with respect to the $W^{1,1}(\Omega )$ topology at $\varphi$ to (1.9). However, in [5, 22] it is observed that this set is only strictly contained in the domain of the $\Gamma$-limit of $AG_\varepsilon$. To identify the asymptotic admissible set, one can exploit the conservation law structure of the eikonal equation (1.8). In particular, suitable notions of entropies (see Remark 3.4 for a short overview) have been exploited to prove compactness properties of the functionals $AG_\varepsilon$ (cf. [5, 26], see also [33] for an approach via the kinetic formulation). Entropies have also been used to define an asymptotic lower bound on the family of functionals $AG_\varepsilon (\cdot , \Omega )$, cf. Remark 3.5. In Section 3 we introduce the functional H, defined in (3.5), which is obtained by taking the supremum of entropy productions over a suitable class of entropies given in Definition 3.1 subject to a normalization constraint. The functional H satisfies the lower bound $H(\nabla \varphi , \Omega ) \leqq \liminf _\varepsilon AG_\varepsilon ( \varphi _\varepsilon , \Omega )$ for $\varphi _\varepsilon \rightarrow \varphi$ in $W^{1,1}(\Omega )$, see (3.12). Moreover, $H(\nabla \varphi , \Omega )$ is given by (1.9) if $\nabla \varphi \in BV$ (cf. Corollary 3.8). As a side note, we mention that the behavior of the sequence of Aviles–Giga functionals is related that of the micromagnetic energies investigated in [43, 50, 51], for which the notion of entropy plays a fundamental role as well.

By carefully adapting to our setting some of the strategies recently exploited to investigate the Aviles–Giga functionals, we can describe the asymptotic behavior of the rescaled $J_1$–$J_2$–$J_3$ energies $H_n \simeq \frac{1}{\delta _n^{3/2} \lambda _n} F_n$. In the main theorem of this paper we prove a compactness and $\Gamma$-convergence result for the functionals $H_n$ that we briefly outline below.

In Theorem 4.1-i) we prove that every sequence $(\chi _n)_n \in L^1_{\mathrm {loc}}(\mathbb {R}^2;\mathbb {R}^2)$ such that

$$\begin{aligned} \sup _n H_n(\chi _n,\Omega ) < +\infty , \end{aligned}$$

is precompact in $L^p_{\mathrm {loc}}(\Omega )$ for every $p \in [1,6)$. Moreover, the limit $\chi$ satisfies $H(\chi , \Omega ) < + \infty$ and, in particular, it solves the eikonal equation in the sense that

$$\begin{aligned} |\chi | = 1 \text { a.e. in } \Omega , \quad \mathrm {curl}(\chi ) = 0 \text { in } \mathcal {D}'(\Omega ). \end{aligned}$$

In Theorem 4.1-ii) we show that the following liminf inequality holds for $H_n$: if $(\chi _n)_n, \chi \in L^1_{\mathrm {loc}}(\mathbb {R}^2;\mathbb {R}^2)$ are such that $\chi _n \rightarrow \chi$ in $L^1_{\mathrm {loc}}(\Omega ;\mathbb {R}^2)$, then

$$\begin{aligned} H(\chi ,\Omega ) \leqq \liminf _{n} H_n(\chi _n,\Omega ). \end{aligned}$$

Finally, assuming the additional scaling assumption $\frac{\delta _n^{5/2}}{\lambda _n} \rightarrow 0$ as $n \rightarrow \infty$, in Theorem 4.1-iii) we prove the following limsup inequality: If $\chi \in L^1_{\mathrm {loc}}(\mathbb {R}^2;\mathbb {R}^2) \cap BV(\Omega ;\mathbb {R}^2)$, then there exists a sequence $(\chi _n)_n \in L^1_{\mathrm {loc}}(\mathbb {R}^2;\mathbb {R}^2)$ such that $\chi _n \rightarrow \chi$ in $L^1(\Omega ;\mathbb {R}^2)$ and

$$\begin{aligned} \limsup _{n} H_n(\chi _n,\Omega ) \leqq H(\chi ,\Omega ). \end{aligned}$$

It is by now well-understood that the variational analysis of discrete-to-continuum problems often does not reduce to the comparison with an analogue continuum model by merely estimating discretization errors. In this sense, compared to the Aviles–Giga functionals, the $J_1$–$J_2$–$J_3$ model features new difficulties, some of which can be recognized by the presence of perturbations of the terms in the energy $H_n$ with respect to those of the Aviles–Giga, see (2.16). In the following we highlight some of the major difficulties in proving our main result. For technical reasons, throughout the paper we will use several different variants of the chirality order parameter, all asymptotically equivalent. Although for the rest of the paper the energy $H_n$ will be defined in terms of the variant denoted by $\chi$, to describe some of the arising difficulties in this introduction, we rewrite it in terms of the parameter $\overline{\chi }= (\overline{\chi }_1, \overline{\chi }_2)$ defined in (2.19) with a slight abuse of notation as follows:

$$\begin{aligned} H_n = \frac{1}{2} \int _\Omega \frac{1}{\varepsilon _n} W^\mathrm {d}\bigg ( \begin{array}{l} \frac{2}{\sqrt{\delta _n}} \sin ( \frac{\sqrt{\delta _n}}{2} \overline{\chi }_1) \\ \frac{2}{\sqrt{\delta _n}} \sin ( \frac{\sqrt{\delta _n}}{2} \overline{\chi }_2) \end{array} \bigg ) + \varepsilon _n\bigg |A^\mathrm {d}\bigg ( \begin{array}{l} \frac{2}{\sqrt{\delta _n}} \sin ( \frac{\sqrt{\delta _n}}{2} \overline{\chi }_1) \\ \frac{2}{\sqrt{\delta _n}} \sin ( \frac{\sqrt{\delta _n}}{2} \overline{\chi }_2) \end{array} \bigg ) \bigg |^2 \, {\mathrm {d}{x}}.\nonumber \\ \end{aligned}$$

(1.10)

In the formula above, $W^\mathrm {d}$ is a discrete approximation of the potential $W(\xi ) = (1-|\xi |^2)^2$ of the Aviles–Giga functionals, and $A^\mathrm {d}$ is an approximation of the divergence operator. More precisely, it is a discrete approximation of the composition of the divergence operator with a $\delta _n$-dependent non-linear perturbation of the identity.

To prove the compactness result Theorem 4.1-i), as a first key step we need to prove a bound on an Aviles–Giga-like energy with unperturbed potential and derivative terms, which we achieve in Proposition 2.6. The crucial step therein is to obtain from the bound on the derivative term featuring $A^\mathrm {d}$ in (1.10) a bound on (a discrete analogue of) the full derivative $\mathrm {D}\overline{\chi }$. This is achieved by recognizing that the derivative term in (1.10) is a non-linear elliptic operator and by employing suitable regularity estimates. Subsequently, in Section 5 we will adapt to our setting the main arguments used in [26] to prove compactness properties of the Aviles–Giga functionals in (1.7).

We prove the liminf inequality in Theorem 4.1-ii) in Section 6. This is achieved by carefully estimating entropy productions in terms of the Aviles–Giga energy as outlined in Remark 6.1, making use of a key observation in [26] that allows us to conveniently rewrite entropy productions. Additionally, in the proof of both the compactness result and the liminf inequality, we have to take care of the fact that $\overline{\chi }$ has possibly non-zero curl, due to the possible formation of vortices in the discrete spin field u. In Lemma 2.3 we prove that the number of such vortex cells can be controlled in terms of the energy. This leads to a rate of convergence of $\mathrm {curl}(\overline{\chi })$ to zero in $L^1$ which we need to use as a replacement of the curl-free condition. The situation we are dealing with here, where the curl concentrates on a controlled number of cells of a certain size, is only natural in the discete. Nevertheless, the question for alternatives to the vanishing curl condition on $\nabla \varphi _\varepsilon$ in the Aviles–Giga functionals $AG_\varepsilon (\varphi _\varepsilon ,\Omega )$ that still lead to the same $\Gamma$-limiting behavior as $\varepsilon \rightarrow 0$ can be asked and may be of interest also in the continuum.

The proof of the limsup inequality in Theorem 4.1-iii) is contained in Section 7. We resort to a technique which has originally been introduced in [46] to prove upper bounds for the Aviles–Giga functionals in (1.7), and has then been generalized to more general singular perturbation functionals in [47]. The latter applies in particular to the energies $AG^\Delta _\varepsilon$ in (1.6). This method has already been successfully applied in [17] to the discrete-to-continuum $\Gamma$-convergence analysis of the simpler $J_1$–$J_3$ model already mentioned in this introduction.

In adapting to our setting the arguments used for the proofs of both the liminf and the limsup inequality a major additional difficulty needs to be overcome. This is due to the fact that in (1.10) the potential term featuring $W^\mathrm {d}$ is, in terms of $\overline{\chi }$, a $\delta _n$-dependent perturbation of the Aviles–Giga potential W with moving wells, i.e., its set of zeros is $\delta _n$-dependent. We stress that in the $\Gamma$-convergence analysis of the Aviles–Giga functionals, dealing with such scale-dependent potentials poses some difficulties even in the continuum case. Due to this issue, we require the additional scaling assumption $\frac{\delta _n^{5/2}}{\lambda _n} \rightarrow 0$ for the proof of the limsup inequality. In contrast, we succeed in proving the liminf inequality without additional assumptions by introducing a class of approximate entropies (cf. Lemma 6.3).

As a final remark, we would like to mention that any rigorous numerical approximation of the Aviles–Giga functionals requires the proof of a $\Gamma$-convergence result of (unperturbed) discretizations of the Aviles–Giga energies, such as the functionals $AG^{\mathrm {d}}_n$ defined in (2.23), as both the discretization parameter $\lambda _n$ and the singular perturbation parameter $\varepsilon _n$ vanish. In the case that $\lambda _n\ll \varepsilon _n$ as $n \rightarrow \infty$, such a result follows as a byproduct of our analysis, cf. Remark 4.5. In fact, for that analysis many of the steps of our proofs can be simplified since several of the aforementioned difficulties due to the non-vanishing curl, the presence of a scale-dependent potential, and the non-linear elliptic derivative term do not take place.

2 Preliminaries and the $J_1$–$J_2$–$J_3$ Model

2.1 Basic notation

Given two vectors $a,b\in \mathbb {R}^m$ we let $a \cdot b$ denote their scalar product. If $a,b\in \mathbb {R}^2$, their cross product is the scalar given by $a {\times }b = a_1 b_2 - a_2 b_1$. As usual, we let $|a| = \sqrt{a \cdot a}$ denote the norm of a. We use the notation $\mathbb {S}^1$ for the unit circle in $\mathbb {R}^2$. Given $a \in \mathbb {R}^N$ and $b \in \mathbb {R}^M$, their tensor product is the matrix $a \otimes b = (a_i b_j)^{i = 1,\dots ,N}_{j = 1,\dots ,M} \in \mathbb {R}^{N \times M}$. Given a vector $\xi = (\xi _1,\xi _2)\in \mathbb {R}^2$, we use the notation $\xi ^\perp := (-\xi _2,\xi _1)$ for the vector obtained by rotating $\xi$ by 90 degrees counterclockwise around the origin.

Given an open set $\Omega \subset \mathbb {R}^d$, we let $\mathcal {M}_b(\Omega ;\mathbb {R}^\ell )$ denote the space of $\mathbb {R}^\ell$-valued Radon measures on $\Omega$ with finite total variation. If $\ell = 1$, i.e., for the space of finite signed Radon measures, we instead use the notation $\mathcal {M}_b(\Omega )$. We define the supremum $\bigvee _{t \in \mathcal {T}} \mu _t$ of a family of non-negative measures $(\mu _t)_{t \in \mathcal {T}} \in \mathcal {M}_b(\Omega )$ (with $\mathcal {T}$ not necessarily countable) by

$$\begin{aligned} \bigvee _{t \in \mathcal {T}} \mu _t (B) := \sup \Big \{ \sum _{t' \in \mathcal {T}'} \mu _{t'}(B_{t'}) : \mathcal {T}' \subset \mathcal {T} \text { finite, } B_{t'} \subset B \text { disjoint Borel sets} \Big \}. \end{aligned}$$

Then $\bigvee _{t \in \mathcal {T}} \mu _t$ is a Borel measure (not necessarily a Radon measure). We recall that if $\mu _t = f_t \mu$ for a non-negative measure $\mu \in \mathcal {M}_b(\Omega )$ and $f_t \geqq 0$ Borel, then $\bigvee _{t \in \mathcal {T}} \mu _t = (\sup _{t \in \mathcal {T}} f_t) \mu$.

Unless specified otherwise, we always let C denote a positive and finite constant that may change at each of its occurences.

2.2 BV functions

In the following we recall some basic facts about BV functions, referring to the book [6] for a comprehensive treatment on the subject. Moreover, we recall the notion of BVG function introduced in [46].

Let $\Omega \subset \mathbb {R}^d$ be an open set. A function $v \in L^1(\Omega ;\mathbb {R}^m)$ is a function of bounded variation if its distributional derivative $\mathrm {D}v$ is a finite matrix-valued Radon measure, i.e., $\mathrm {D}v \in \mathcal {M}_b(\Omega ;\mathbb {R}^{m {\times }d})$.

The distributional derivative $\mathrm {D}v \in \mathcal {M}_b(\Omega ;\mathbb {R}^{m {\times }d})$ of a function $v \in BV(\Omega ;\mathbb {R}^m)$ can be decomposed in the sum of three mutually singular matrix-valued measures

(2.1)

where $\mathcal {L}^d$ is the Lebesgue measure and $\mathcal {H}^{d-1}$ is the $(d-1)$-dimensional Hausdorff measure; $\nabla v \in L^1(\Omega ;\mathbb {R}^{m {\times }d})$ is the approximate gradient of v; $\mathrm {D}^c v$ is the so-called Cantor part of the derivative satisfying $\mathrm {D}^c v(B) = 0$ for every Borel set B with $\mathcal {H}^{n-1}(B) < \infty$; $J_v$ denotes the jump set of v, $\nu _v$ denotes the direction of the jump, $[v] = (v^+ - v^-)$, and $v^+$ and $v^-$ denote the one-sided approximate limits of v on $J_v$. These are defined for a general $w \in L^1_{\mathrm {loc}}(\Omega ;\mathbb {R}^m)$ as follows (cf. for example [6, Definition 3.67]): $J_w$ is the set of points $x \in \Omega$ such that there exist $a,b \in \mathbb {R}^m$, $a \ne b$, and $\nu \in \mathbb {S}^1$ such that

$$\begin{aligned}&\lim _{r \rightarrow 0} \frac{1}{r^2} \int _{B^{+}_r(x,\nu )} |w(y) - a| \, {\mathrm {d}{y}} = 0, \nonumber \\&\quad \lim _{r \rightarrow 0} \frac{1}{r^2} \int _{B^{-}_r(x,\nu )} |w(y) - b| \, {\mathrm {d}{y}} = 0 \end{aligned}$$

(2.2)

with $B^{\pm }_r(x,\nu ) = \{ y \in B_r(x)\ : \ \pm (y-x) \cdot \nu > 0 \}$. The triple $(a,b,\nu )$ is unique up to the change to $(b,a,-\nu )$ and referred to as $(w^+(x),w^-(x),\nu _w(x))$. We let $[w](x) := w^+(x) - w^-(x)$.

We recall that every function $v \in BV(\Omega ; \mathbb {R}^m)$ is approximately continuous at $\mathcal {H}^{d-1}$-a.e. point $x \in \Omega \setminus J_v$, in the sense that

$$\begin{aligned} \lim _{r \rightarrow 0} \frac{1}{r^d} \int _{B_r(x)} |v(y) - \xi | \, {\mathrm {d}{y}} = 0 \end{aligned}$$

for some $\xi \in \mathbb {R}^m$. The point $\xi$ is called the approximate limit of v at x and coincides with v(x) for $\mathcal {L}^{d}$-a.e. x.

Let us furthermore recall the Vol’pert chain rule: Let $v \in BV(\Omega ;\mathbb {R}^m)$ and let $\Phi \in C^1(\mathbb {R}^m;\mathbb {R}^\ell )$ be Lipschitz. If $\mathcal {L}^d(\Omega ) = + \infty$, assume moreover that $\Phi (0) = 0$. Then, $\Phi \circ v \in BV(\Omega ;\mathbb {R}^\ell )$ and

Note carefully that here the term $\mathrm {D}\Phi (v)$ has to be understood as the function defined up to an $\mathcal {H}^{d-1}$-null set on $\Omega \setminus J_v$ by $\mathrm {D}\Phi (v)(x) := \mathrm {D}\Phi (\xi )$, where $\xi$ is the approximate limit of v at x.

Finally, we recall the space $BVG(\Omega )$ introduced in [46]. This is defined by

$$\begin{aligned} BVG(\Omega ) := \big \{ \varphi \in W^{1,\infty }(\Omega ) \ : \ \nabla \varphi \in BV(\Omega ;\mathbb {R}^2) \}. \end{aligned}$$

In [46], the author proves a convenient extension result for functions in $BVG(\Omega )$ under suitable conditions on the regularity of the set $\Omega$. A bounded, open set $\Omega \subset \mathbb {R}^d$ is called a BVG domain if $\Omega$ can be described locally at its boundary as the epigraph of a BVG function $\mathbb {R}^{d-1} \rightarrow \mathbb {R}$ with respect to a suitable choice of the axes, i.e., if every $x \in \partial \Omega$ has a neighborhood $U_x \subset \mathbb {R}^d$ such that there exists a function $\psi _x \in BVG(\mathbb {R}^{d-1})$ and a rigid motion $R_x :\mathbb {R}^d \rightarrow \mathbb {R}^d$ satisfying

$$\begin{aligned} R_x ( \Omega \cap U_x ) = \{ y = (y_1,y') \in \mathbb {R}{\times }\mathbb {R}^{d-1} \ : \ y_1 > \psi _x(y') \} \cap R_x (U_x). \end{aligned}$$

Every BVG domain is an extension domain for BVG functions in the following sense.

Proposition 2.1

(Proposition 4.1 in [46]) Let $\Omega$ be a BVG domain. Then for every $\varphi \in BVG(\Omega )$ there exists $\overline{\varphi }\in BVG(\mathbb {R}^d)$ such that $\overline{\varphi }= \varphi$ in $\Omega$ and $|\mathrm {D}\nabla \overline{\varphi }|(\partial \Omega ) = 0$.

2.3 Jumps of functions with vanishing curl

We recall here how the curl-free constraint of a vector field enforces a relation between the geometry of its jump set and its one-sided approximate limits on both sides of the jump. For simplicity, we restrict to vector fields in dimension $d = 2$. In the following, $\Omega$ is an open subset of $\mathbb {R}^2$.

Given a vector field $v \in L^1_{\mathrm {loc}}(\Omega ;\mathbb {R}^2)$, we define its (distributional) curl by $\mathrm {curl}(v) := \partial _1 v_2 - \partial _2 v_1$, the partial derivatives being taken in the distributional sense.

If $v \in BV(\Omega ;\mathbb {R}^2)$, it is clear from (2.1) that $\mathrm {curl}(v) = 0$ implies that

and, as a consequence, [v] is parallel to $\nu _v$ at $\mathcal {H}^1$-a.e. point in $J_v$.

If $v \in L^1_{\mathrm {loc}}(\Omega ;\mathbb {R}^2)$ satisfies $\mathrm {curl}(v) = 0$, it can be observed that still [v] is parallel to $\nu _v$, and in fact this holds everywhere on $J_v$. Indeed, being $\mathrm {curl}(v) = 0$, the same is true for the rescaled functions $v^{x,r}(y) := v(x+ry)$ for $x \in \Omega$ and $r > 0$. Taking $x \in J_v$ and letting $r \rightarrow 0$, by (2.2) we get that $v^{x,r}$ converge in $L^1(B_1(0))$ to the pure jump function

$$\begin{aligned} j_{v^+(x),v^-(x)}^{\nu _v(x)} :y \mapsto {\left\{ \begin{array}{ll} v^+(x) &{} \text {if } y \cdot \nu _v(x) > 0, \\ v^-(x) &{} \text {if } y \cdot \nu _v(x) < 0. \end{array}\right. } \end{aligned}$$

As a consequence we get that $\mathrm {curl}(j_{v^+(x),v^-(x)}^{\nu _v(x)}) = 0$. Since $j_{v^+(x),v^-(x)}^{\nu _v(x)}$ is a BV vector field, this yields that [v](x) is parallel to $\nu _v(x)$.

2.4 Discrete functions

We introduce here the notation used for functions defined on a square lattice in $\mathbb {R}^2$. For the whole paper, $\lambda _n$ denotes a sequence of positive lattice spacings that converges to zero. Given $i,j \in \mathbb {Z}$, we define the half-open square $Q_{\lambda _n}(i,j)$ with left-bottom corner in $(\lambda _ni, \lambda _nj)$ by $Q_{\lambda _n}(i,j) := (\lambda _ni, \lambda _nj) + [0,\lambda _n)^2$ . We refer to $Q_{\lambda _n}(i,j)$ as a cell of the lattice $\lambda _n\mathbb {Z}^2$. For a given set S, we introduce the class of functions with values in S which are piecewise constant on the cells of the lattice $\lambda _n\mathbb {Z}^2$:

$$\begin{aligned} \mathcal {PC}_{\lambda _n}(S) := \{ v :\mathbb {R}^2 \rightarrow S \ : \ v(x) = v(\lambda _ni, \lambda _nj) \text { for } x \in Q_{\lambda _n}(i,j)\}. \end{aligned}$$

With a slight abuse of notation, we will always identify a function $v \in \mathcal {PC}_{\lambda _n}(S)$ with the function defined on the points of the lattice $\mathbb {Z}^2$ given by $(i,j) \mapsto v^{i,j} := v(\lambda _ni, \lambda _nj)$ for $(i,j) \in \mathbb {Z}^2$. Conversely, given values $v^{i,j} \in S$ for $(i,j) \in \mathbb {Z}^2$, we define $v \in \mathcal {PC}_{\lambda _n}(S)$ by $v(x) := v^{i,j}$ for $x \in Q_{\lambda _n}(i,j)$. Given a sequence $v_n \in \mathcal {PC}_{\lambda _n}(\mathbb {R}^m)$, we use the notation $v_{k,n}^{i,j}$ to refer to the k-th component of $v_n^{i,j}$.

Given $v \in \mathcal {PC}_{\lambda _n}(\mathbb {R}^m)$, we define its discrete partial derivatives $\partial ^{\mathrm {d}}_1 v , \partial ^{\mathrm {d}}_2 v \in \mathcal {PC}_{\lambda _n}(\mathbb {R}^m)$ by $\partial ^{\mathrm {d}}_1 v^{i,j} := \frac{1}{\lambda _n}(v^{i+1,j} - v^{i,j})$ and $\partial ^{\mathrm {d}}_2 v^{i,j} := \frac{1}{\lambda _n}(v^{i,j+1} - v^{i,j})$. Using these discrete derivatives, we have analogues of any differential operator in the discrete. In particular, we define $\mathrm {D}^{\mathrm {d}}v \in \mathcal {PC}_{\lambda _n}(\mathbb {R}^{m \times 2})$ to be the matrix whose k-th column is given by $\partial ^{\mathrm {d}}_k v$. If $m = 1$, we will often interpret $\mathrm {D}^{\mathrm {d}}v^{i,j}$ instead as a vector in $\mathbb {R}^2$. Moreover, if $m = 2$, we define $\mathrm {div}^{\mathrm {d}}(v) \in \mathcal {PC}_{\lambda _n}(\mathbb {R})$ and $\mathrm {curl}^{\mathrm {d}}(v) \in \mathcal {PC}_{\lambda _n}(\mathbb {R})$ by

$$\begin{aligned} \mathrm {div}^{\mathrm {d}}(v)^{i,j} := \mathrm {tr} (\mathrm {D}^{\mathrm {d}}v^{i,j}) = \partial ^{\mathrm {d}}_1 v_1^{i,j} + \partial ^{\mathrm {d}}_2 v_2^{i,j} \quad \text {and} \quad \mathrm {curl}^{\mathrm {d}}(v)^{i,j} := \partial ^{\mathrm {d}}_1 v_2^{i,j} - \partial ^{\mathrm {d}}_2 v_1^{i,j} \end{aligned}$$

and call them the discrete divergence and the discrete curl of v, respectively. It is to be noted that in some contexts the proper discrete analogue of the Laplacian $\Delta$ of a field $v \in \mathcal {PC}_{\lambda _n}(\mathbb {R})$ is given by

$$\begin{aligned} \Delta _{\mathrm {s}}^{\mathrm {d}}v^{i,j} := \partial ^{\mathrm {d}}_{11} v^{i-1,j} + \partial ^{\mathrm {d}}_{22} v^{i,j-1}, \end{aligned}$$

(2.3)

i.e., suitable shifts in the lattice points are needed. To reflect this fact we add to our notation the subscript $\mathrm {s}$ which stands for “shifted".

Next, we mention here a specific type of interpolation, which we shall use several times throughout the paper, mainly to relate the discrete divergence of a discrete vector field to its distributional divergence. For any $v \in \mathcal {PC}_{\lambda _n}(\mathbb {R}^2)$ we define ${\mathcal {I}} v :\mathbb {R}^2 \rightarrow \mathbb {R}^2$ as follows: Given any cell $Q_{\lambda _n}(i,j)$ of the lattice $\lambda _n\mathbb {Z}^2$ and any $x \in Q_{\lambda _n}(i,j)$, we write $x = \lambda _n(i,j) + \lambda _ny$, where $y= (y_1,y_2) \in [0,1)^2$. We set

$$\begin{aligned} \mathcal {I} v(x) := \begin{pmatrix} (1-y_1) v_1^{i,j} + y_1 v_1^{i+1,j} \\[3pt] (1-y_2) v_2^{i,j} + y_2 v_2^{i,j+1} \end{pmatrix}. \end{aligned}$$

(2.4)

We observe that $\mathrm {div}( \mathcal {I}v) = \mathrm {div}^{\mathrm {d}}(v)$ in the sense of distributions. In particular, $\mathrm {curl}^{\mathrm {d}}(v) = - \mathrm {div}(\mathcal {I}(v^\perp ))$. Moreover, we note that

$$\begin{aligned} |\mathcal {I} v (x) - v(x)| = \left| \begin{pmatrix} y_1 \lambda _n\partial ^{\mathrm {d}}_1 v_1^{i,j} \\[3pt] y_2 \lambda _n\partial ^{\mathrm {d}}_2 v_2^{i,j} \end{pmatrix} \right| \leqq C \lambda _n|\mathrm {D}^{\mathrm {d}}v(x)| \, \end{aligned}$$

(2.5)

for $x = \lambda _n(i,j) + \lambda _ny \in Q_{\lambda _n}(i,j)$.

The energy of the model (cf. Section 2.5 below) is defined on spin fields $u \in \mathcal {PC}_{\lambda _n}(\mathbb {S}^1)$. To every such u we associate the oriented angles $\theta ^{\mathrm {hor}}(u)$, $\theta ^{\mathrm {ver}}(u) \in \mathcal {PC}_{\lambda _n}([-\pi ,\pi ))$ between adjacent spins by

$$\begin{aligned} \begin{aligned} (\theta ^{\mathrm {hor}}(u))^{i,j}&:= \mathrm {sign}( u^{i,j} {\times }u^{i+1,j} ) \arccos ( u^{i,j} \cdot u^{i+1,j} ), \\ (\theta ^{\mathrm {ver}}(u))^{i,j}&:= \mathrm {sign}( u^{i,j} {\times }u^{i,j+1} ) \arccos ( u^{i,j} \cdot u^{i,j+1} ), \end{aligned} \end{aligned}$$

(2.6)

where we used the convention $\mathrm {sign}(0) = -1$. We shall often drop the dependence on u as it will be clear from the context and for shortness we adopt the notation $\theta ^{\mathrm {hor}}_n$ and $\theta ^{\mathrm {ver}}_n$ for the angles associated to $u_n$.

2.5 Derivation of the energy model

The main subject of our study will be the sequence of functionals $H_n$ which we define in Section 2.6 below. We show here how these are derived from the energies $E_n$ in (1.1).

We start by showing how the energy $E_n$ in (1.1) can be written in terms of the energy $F_n$ in (1.2). In the following, we let the sums run over indices (i, j) such that $(\lambda _ni, \lambda _nj)$ belongs to a fixed set $\Omega$. We shall specify later the precise assumptions on $\Omega$, as now we present a formal computation. We split the terms in the sum involving $\alpha _n$ as follows:

$$\begin{aligned} \begin{aligned} E_n(u)&= - \alpha _n\lambda _n^2 \sum _{(i,j)} \Big ( u^{i,j} \cdot u^{i+1,j} + u^{i,j} \cdot u^{i,j+1} \Big ) \\&\quad + \beta _n\lambda _n^2 \sum _{(i,j)} \Big ( u^{i,j} \cdot u^{i+1,j+1} + u^{i,j} \cdot u^{i-1,j+1} \Big ) \\&\quad + \lambda _n^2 \sum _{(i,j)} \Big ( u^{i,j} \cdot u^{i+2,j} + u^{i,j} \cdot u^{i,j+2} \Big ) \\&= \lambda _n^2 \sum _{(i,j)} \Big ( \Big (-\frac{\beta _n\alpha _n}{\beta _n+2}- \frac{2 \alpha _n}{\beta _n+2} \Big ) u^{i,j} \cdot u^{i+1,j} \\&\quad + \Big (- \frac{\beta _n\alpha _n}{\beta _n+2}- \frac{2 \alpha _n}{\beta _n+2} \Big ) u^{i,j} \cdot u^{i,j+1}\Big ) \\&\quad + \beta _n\lambda _n^2 \sum _{(i,j)} \Big ( u^{i,j} \cdot u^{i+1,j+1} + u^{i,j} \cdot u^{i-1,j+1} \Big ) \\&\quad + \lambda _n^2 \sum _{(i,j)} \Big ( u^{i,j} \cdot u^{i+2,j} + u^{i,j} \cdot u^{i,j+2} \Big ). \end{aligned} \end{aligned}$$

Then we shift coordinates: in $u^{i,j} \cdot u^{i+1,j}$ to get $\tfrac{1}{2} u^{i,j} \cdot u^{i+1,j}$ and $\tfrac{1}{2} u^{i-1,j} \cdot u^{i,j}$; in $u^{i,j} \cdot u^{i,j+1}$ to get $\tfrac{1}{2} u^{i,j} \cdot u^{i,j+1}$ and $\tfrac{1}{2} u^{i,j-1} \cdot u^{i,j}$; in $u^{i,j} \cdot u^{i-1,j+1}$ to get $\tfrac{1}{2}u^{i+1,j} \cdot u^{i,j+1}$ and $\tfrac{1}{2} u^{i-1,j} \cdot u^{i,j-1}$; in $u^{i,j} \cdot u^{i+1,j+1}$ to get $\tfrac{1}{2}u^{i+1,j} \cdot u^{i,j-1}$ and $\tfrac{1}{2}u^{i-1,j} \cdot u^{i,j+1}$; in $u^{i,j}\cdot u^{i+2,j}$ to get $u^{i-1,j}\cdot u^{i+1,j}$; in $u^{i,j}\cdot u^{i,j+2}$ to get $u^{i,j-1}\cdot u^{i,j+1}$.

The shifting procedure above may produce energy errors when applied to points $(\lambda _ni,\lambda _nj)$ close to the boundary of $\Omega$. For instance a pair (i, j) such that $(\lambda _ni,\lambda _nj)\in \Omega$ could be transformed into a new shifted pair $(i',j')$ such that $(\lambda _ni',\lambda _nj')\not \in \Omega$ and, as such, it could no more be an element of the sum. Letting $B_n$ denote these errors, we obtain

$$\begin{aligned} E_n(u)&= \lambda _n^2 \sum _{(i,j)} \Big ( - \frac{\alpha _n}{\beta _n+2} u^{i,j} \cdot u^{i+1,j} + u^{i-1,j} \cdot u^{i+1,j} - \frac{\alpha _n}{\beta _n+2} u^{i-1,j} \cdot u^{i,j} \\&\quad - \frac{\alpha _n}{\beta _n+2} u^{i,j} \cdot u^{i,j+1} + u^{i,j-1} \cdot u^{i,j+1} - \frac{ \alpha _n}{\beta _n+2} u^{i,j-1} \cdot u^{i,j} \Big ) \\&\quad + \frac{\beta _n}{2} \lambda _n^2 \sum _{(i,j)} \Big ( u^{i+1,j} \cdot u^{i,j+1} - \frac{\alpha _n}{\beta _n+2} u^{i,j} \cdot u^{i+1,j} + u^{i+1,j} \cdot u^{i,j-1} \\&\quad - \frac{\alpha _n}{\beta _n+2} u^{i,j} \cdot u^{i,j+1} - \frac{\alpha _n}{\beta _n+2} u^{i,j-1} \cdot u^{i,j} \\&\quad + u^{i-1,j} \cdot u^{i,j+1} - \frac{\alpha _n}{\beta _n+2} u^{i-1,j} \cdot u^{i,j} + u^{i-1,j} \cdot u^{i,j-1} \Big ) + B_n. \end{aligned}$$

By reorganizing the terms we get that

$$\begin{aligned} E_n(u)&= \frac{1}{2} \lambda _n^2 \sum _{(i,j)} \Big | u^{i+1,j} - \frac{\alpha _n}{\beta _n+2} u^{i,j} + u^{i-1,j} \Big |^2 \\&\quad + \Big | u^{i,j+1} - \frac{\alpha _n}{\beta _n+2} u^{i,j} + u^{i,j-1} \Big |^2 \\&\quad + \frac{\beta _n}{2} \lambda _n^2 \sum _{(i,j)} \Big ( u^{i+1,j} - \frac{\alpha _n}{\beta _n+2} u^{i,j} + u^{i-1,j} \Big )\\&\quad \cdot \Big ( u^{i,j+1} - \frac{\alpha _n}{\beta _n+2} u^{i,j} + u^{i,j-1} \Big ) \\&\quad - \lambda _n^2 \sum _{(i,j)} 2 + \frac{2\alpha _n^2}{2(\beta _n+2)^2} + \frac{\alpha _n^2 \beta _n}{2(\beta _n+2)^2} + B_n\\&= F_n(u) - \lambda _n^2 \sum _{(i,j)} \Big ( \frac{\alpha _n^2 }{2(\beta _n+2)} + 2 \Big ) + B_n, \end{aligned}$$

where $F_n$ is given by (1.2). As here we are not interested in the energy due to boundary layers, we shall neglect the error term $B_n$. Removing from $E_n$ the bulk energy $-\lambda _n^2 \sum _{(i,j)} \big ( \frac{\alpha _n^2 }{2(\beta _n+2)} + 2 \big )$ corresponding to the energy of the ground states, we are led to study the energy $F_n$.

As explained in the introduction, the main results in this paper concern the case where $\alpha _n< 8$, $\alpha _n\rightarrow 8$, and $\beta _n\equiv 2$. In that case the energy $F_n$ reads as

$$\begin{aligned} F_n(u) = \frac{1}{2} \lambda _n^2 \sum _{(i,j)} \Big | u^{i+1,j} + u^{i-1,j} + u^{i,j+1} + u^{i,j-1} - \frac{\alpha _n}{2} u^{i,j} \Big |^2. \end{aligned}$$

(2.7)

We find it convenient to parametrize the convergence $\alpha _n\rightarrow 8$ by introducing the positive sequence $\delta _n:= 4 - \frac{\alpha _n}{2}$ such that $\delta _n\rightarrow 0$.

Next, we introduce an order parameter $\chi (u)$ representing the chirality of the spin field u and we express the above energy in terms of this parameter. A rescaling will then lead to the energies $H_n$. Due to technical reasons we need to work with several variants of the chirality parameter. More specifically, we define

$$\begin{aligned} \begin{alignedat}{2} \chi (u)^{i,j}&:= \big (\chi _1(u)^{i,j}, \chi _2(u)^{i,j} \big ) , \quad&{\widetilde{\chi }}(u)^{i,j} := \big ({\widetilde{\chi }}_1(u)^{i,j}, {\widetilde{\chi }}_2(u)^{i,j} \big ), \text { where} \\ \chi _1(u)^{i,j}&:= \frac{2}{\sqrt{\delta _n}}\sin \Big (\frac{(\theta ^{\mathrm {hor}})^{i,j}}{2}\Big ), \quad&{\widetilde{\chi }}_1(u)^{i,j} := \frac{1}{\sqrt{\delta _n}} \sin \big ((\theta ^{\mathrm {hor}})^{i,j}\big ), \\ \chi _2(u)^{i,j}&:= \frac{2}{\sqrt{\delta _n}}\sin \Big (\frac{(\theta ^{\mathrm {ver}})^{i,j}}{2}\Big ), \quad&{\widetilde{\chi }}_2(u)^{i,j} := \frac{1}{\sqrt{\delta _n}}\sin \big ( (\theta ^{\mathrm {ver}})^{i,j} \big ), \end{alignedat} \end{aligned}$$

(2.8)

where $\theta ^{\mathrm {hor}}= \theta ^{\mathrm {hor}}(u)$ and $\theta ^{\mathrm {ver}}=\theta ^{\mathrm {ver}}(u)$ are given by (2.6). A third variant $\overline{\chi }(u)$ will be introduced in (2.19) below. In our notation we shall often drop the dependence on u as it will be clear from the context. In addition, given a sequence of spin fields $u_n$, we will write $\chi _n, {\widetilde{\chi }}_n$ in place of $\chi (u_n), {\widetilde{\chi }}(u_n)$, respectively. Note that ${\widetilde{\chi }}$ can be written as a function of $\chi$, e.g., ${\widetilde{\chi }}_1^{i,j} = \frac{1}{\sqrt{\delta _n}} \sin \big ( 2 \arcsin \big (\frac{\sqrt{\delta _n}}{2} \chi _1^{i,j} \big ) \big )$. Since $\delta _n\rightarrow 0$, the reader can formally assume that $\chi \simeq {\widetilde{\chi }}$ as $n \rightarrow \infty$ to ease the reading of the statements.

Given (i, j), we rewrite the corresponding contribution to the energy in (2.7) in terms of $\chi (u)$. We observe that the $\mathbb {S}^1$-symmetry of the energy allows us to assume, without loss of generality, that $u^{i,j} = \exp (\iota 0) = (1,0)$. Here and in the following, we interpret vectors in $\mathbb {S}^1$ as complex numbers via the relation $(\cos \theta , \sin \theta ) = e^{\iota \theta }$, where $\iota$ is the imaginary unit. As a consequence, the spins appearing in (2.7) can be rewritten in terms of the relative angles as

$$\begin{aligned} u^{i+1,j}&= e^{\iota (\theta ^{\mathrm {hor}})^{i,j}} , \ \ u^{i-1,j}&= e^{-\iota (\theta ^{\mathrm {hor}})^{i-1,j}} , \\ u^{i,j+1}&= e^{\iota (\theta ^{\mathrm {ver}})^{i,j}} , \ \ u^{i,j-1}&= e^{-\iota (\theta ^{\mathrm {ver}})^{i,j-1}} . \end{aligned}$$

We rewrite the energy $F_n(u)$ in terms of $\theta ^{\mathrm {hor}}$ and $\theta ^{\mathrm {ver}}$ as follows:

$$\begin{aligned} \begin{aligned} F_n(u)&= \frac{1}{2} \lambda _n^2 \sum _{(i,j)} \Big | e^{\iota (\theta ^{\mathrm {hor}})^{i,j}} + e^{-\iota (\theta ^{\mathrm {hor}})^{i-1,j}} + e^{\iota (\theta ^{\mathrm {ver}})^{i,j}} + e^{-\iota (\theta ^{\mathrm {ver}})^{i,j-1}} - \frac{\alpha _n}{2} (1,0) \Big |^2 \\&= \frac{1}{2} \lambda _n^2 \sum _{(i,j)} \Big (\cos (\theta ^{\mathrm {hor}})^{i,j} +\cos (\theta ^{\mathrm {hor}})^{i-1,j} \\&\quad + \cos (\theta ^{\mathrm {ver}})^{i,j} + \cos (\theta ^{\mathrm {ver}})^{i,j-1} - \frac{\alpha _n}{2} \Big )^{\! 2} \\&\quad + \frac{1}{2} \lambda _n^2 \sum _{(i,j)} \Big ( \sin (\theta ^{\mathrm {hor}})^{i,j} - \sin (\theta ^{\mathrm {hor}})^{i-1,j} + \sin (\theta ^{\mathrm {ver}})^{i,j} -\sin (\theta ^{\mathrm {ver}})^{i,j-1} \Big )^{\! 2} . \end{aligned} \end{aligned}$$

(2.9)

Using the trigonometric identity $\cos (\theta ) = 1-2\sin ^2\big (\frac{\theta }{2}\big )$ and recalling that $\delta _n= 4 - \frac{\alpha _n}{2}$, we get that

$$\begin{aligned} \begin{aligned} F_n(u)&= \frac{1}{2} \lambda _n^2 \sum _{(i,j)} \Big ( \delta _n- 2 \sin ^2\Big (\frac{(\theta ^{\mathrm {hor}})^{i,j}}{2}\Big ) - 2 \sin ^2\Big (\frac{(\theta ^{\mathrm {hor}})^{i-1,j}}{2}\Big ) \\&\quad - 2 \sin ^2\Big (\frac{(\theta ^{\mathrm {ver}})^{i,j}}{2}\Big ) - 2 \sin ^2\Big (\frac{(\theta ^{\mathrm {ver}})^{i,j-1}}{2}\Big ) \Big )^{\! 2} \\&\quad + \frac{1}{2} \lambda _n^2 \sum _{(i,j)} \Big (\lambda _n\partial ^{\mathrm {d}}_1 \sin \big ( (\theta ^{\mathrm {hor}})^{i-1,j} \big ) + \lambda _n\partial ^{\mathrm {d}}_2 \sin \big ( (\theta ^{\mathrm {ver}})^{i,j-1} \big )\Big )^{\!2} . \end{aligned} \end{aligned}$$

(2.10)

Finally, using the definition of $\chi$, we obtain that

$$\begin{aligned} \begin{aligned} F_n(u)&= \frac{1}{2} \lambda _n^2 \sum _{(i,j)} \frac{\delta _n^2}{4}\Big (2 - |\chi _1^{i,j}|^2 - |\chi _1^{i-1,j}|^2 - |\chi _2^{i,j}|^2 - |\chi _2^{i,j-1}|^2 \Big )^{\! 2} \\&\quad + \delta _n\lambda _n^2 \Big |\partial ^{\mathrm {d}}_1 {\widetilde{\chi }}_1^{i-1,j} + \partial ^{\mathrm {d}}_2 {\widetilde{\chi }}_2^{i,j-1} \Big |^{ 2} \\&= \frac{\delta _n^{3/2} \lambda _n}{2} \lambda _n^2 \sum _{(i,j)} \frac{\sqrt{\delta _n}}{\lambda _n} W^\mathrm {d}(\chi )^{i,j} + \frac{\lambda _n}{\sqrt{\delta _n}} |A^\mathrm {d}(\chi )^{i,j}|^2 \\&= \frac{\delta _n^{3/2} \lambda _n}{2} \int _{\Omega _{\lambda _n}} \frac{1}{\varepsilon _n} W^\mathrm {d}(\chi ) + \varepsilon _n|A^\mathrm {d}(\chi )|^2 \, {\mathrm {d}{x}}. \end{aligned} \end{aligned}$$

(2.11)

In the above formula, we let $\Omega _{\lambda _n}$ denote the union of cells of the lattice appearing in the sum and we define $\varepsilon _n:= \frac{\lambda _n}{\sqrt{\delta _n}}$. Moreover, we have associated to $\chi$ the piecewise constant functions $W^\mathrm {d}(\chi )$, $A^\mathrm {d}(\chi ) \in \mathcal {PC}_{\lambda _n}(\mathbb {R})$ defined by

$$\begin{aligned}&W^\mathrm {d}(\chi )^{i,j} := \frac{1}{4}\Big (2 - |\chi _1^{i,j}|^2 - |\chi _1^{i-1,j}|^2 - |\chi _2^{i,j}|^2 - |\chi _2^{i,j-1}|^2 \Big )^2 , \nonumber \\&\quad A^\mathrm {d}(\chi )^{i,j} := \partial ^{\mathrm {d}}_1 {\widetilde{\chi }}_1^{i-1,j} + \partial ^{\mathrm {d}}_2 {\widetilde{\chi }}_2^{i,j-1} , \end{aligned}$$

(2.12)

with $\chi$, ${\widetilde{\chi }}$ given by the relations (2.8) and recalling that ${\widetilde{\chi }}$ can be written as a function of $\chi$. The integral in the right-hand side of (2.11) defines the functional we are interested in in this paper. However, working with the integral on $\Omega _{\lambda _n}$ instead of $\Omega$ gives rise to minor technical issues, which are only tedious to fix. For this reason, in this paper we study directly the integral functional on $\Omega$, which we define precisely in Section 2.6 below.

2.6 Assumptions on the model, the energies $H_n$, and the Aviles–Giga functionals

Throughout the paper we assume that $\lambda _n, \delta _n$ are two sequences of positive real numbers that converge to zero such that

$$\begin{aligned} \varepsilon _n:= \frac{\lambda _n}{\sqrt{\delta _n}} \rightarrow 0 \quad \text {as } n \rightarrow \infty \, . \end{aligned}$$

(2.13)

In particular, we have that $\lambda _n\ll \varepsilon _n$ as $n \rightarrow \infty$.

Our main result is valid whenever the domain $\Omega$ belongs to the class of admissible domains defined by

$$\begin{aligned} \mathcal {A}_0 := \{ \Omega \subset \mathbb {R}^2 \ : \ \Omega \text { is an open, bounded, simply connected},\, BVG\text { domain}\}.\nonumber \\ \end{aligned}$$

(2.14)

We recall that simply connected sets are by definition connected. Since parts of our results remain true under more general assumptions on $\Omega$ (cf. Remark 4.3), let us also introduce

$$\begin{aligned} \mathcal {A}:= \{ \Omega \subset \mathbb {R}^2 \ : \ \Omega \text { is an open and bounded set}\}. \end{aligned}$$

(2.15)

In the rest of this section, $\Omega$ is always a domain in $\mathcal {A}$. The theorems in this paper will be stated for the functionals $H_n :L^1_{\mathrm {loc}}(\mathbb {R}^2;\mathbb {R}^2) {\times }\mathcal {A}\rightarrow [0,+\infty ]$ defined by

$$\begin{aligned} H_n(\chi ,\Omega ) := \frac{1}{2} \int _\Omega \frac{1}{\varepsilon _n} W^\mathrm {d}(\chi ) + \varepsilon _n|A^\mathrm {d}(\chi )|^2 \, {\mathrm {d}{x}}, \end{aligned}$$

(2.16)

if $\chi = \chi (u)$ as in (2.8) for some $u \in \mathcal {PC}_{\lambda _n}(\mathbb {S}^1)$, and $H_n$ extended to $+\infty$ otherwise. As a conclusion of Section 2.5, we have established in which sense

$$\begin{aligned} \frac{1}{\delta _n^{3/2}\lambda _n} (E_n(u)-\min E_n) \sim H_n(\chi ,\Omega ) . \end{aligned}$$

(2.17)

As remarked in the introduction, the functionals $H_n$ are related to the Aviles–Giga functionals. Indeed, let us note that $W^\mathrm {d}$ is a discrete approximation of the potential

$$\begin{aligned} W :\mathbb {R}^2 \rightarrow [0, + \infty ), \quad W(\xi ) := (1-|\xi |^2)^2 \end{aligned}$$

(2.18)

with suitable shifts in the discrete variable. Moreover, let us note that in a similar way, we have that $A^\mathrm {d}(\chi ) \simeq \mathrm {div}^{\mathrm {d}}({\widetilde{\chi }})$. Since ${\widetilde{\chi }} \simeq \chi$ for large n, the functionals $H_n$ resemble a discretization of the functionals $\chi \mapsto \frac{1}{2} \int _{\Omega } \frac{1}{\varepsilon _n} W(\chi ) + \varepsilon _n|\mathrm {div}(\chi )|^2 \, {\mathrm {d}{x}}$. In Remark 2.5 below we show how $\mathrm {curl}(\chi ) \simeq 0$ which fully establishes the relation of $H_n$ to the Aviles–Giga-like functional $AG^\Delta _{\varepsilon _n}$ in (1.6), which in turn is related to the classical Aviles–Giga functional $AG_{\varepsilon _n}$ in (1.7).

In what follows, we will explore this relation in more detail and to this end, as well as for later use, prove several a priori estimates on $\chi$ that can be obtained from the energy bound $H_n(\chi , \Omega ) \leqq C$.

Remark 2.2

The potential part in $H_n$ provides $L^4$ bounds on the variable $\chi$. More precisely, if $\sup _n H_n(\chi _n,\Omega ) < + \infty$, then $\sup _n \Vert \chi _n\Vert _{L^4(\Omega )} < + \infty$. Indeed, using Young’s inequality, we have that $(2-a)^2 \geqq \frac{1}{2} a^2 - 4$ for $a = |\chi _{1,n}^{i,j}|^2 + |\chi _{1,n}^{i-1,j}|^2 + |\chi _{2,n}^{i,j}|^2 + |\chi _{2,n}^{i,j-1}|^2$. As a consequence,

$$\begin{aligned} C\geqq & {} \frac{1}{4 \varepsilon _n} \int _{\Omega } \frac{1}{2} \big ( |\chi _{1,n}^{i,j}|^2 + |\chi _{1,n}^{i-1,j}|^2 + |\chi _{2,n}^{i,j}|^2 + |\chi _{2,n}^{i,j-1}|^2 \big )^2 - 4 \, {\mathrm {d}{x}}\\\geqq & {} \frac{1}{4 \varepsilon _n} \int _{\Omega } \frac{1}{2} |\chi _n|^4 - 4 \, {\mathrm {d}{x}}. \end{aligned}$$

This bound can be improved by additionally exploiting the derivative part in $H_n$ as explained in detail below in Proposition 2.7.

Using the potential part of $H_n$, in the following lemma we count the number of cells where the angles between adjacent spins defined in (2.6) are far from 0. This counting argument will be often put to use throughout the paper.

Lemma 2.3

Assume that $\sup _n H_n(\chi _n,\Omega ) < + \infty$. Then for every $t \in (0,+\infty )$ there exists $C(t) \in (0,+\infty )$ such that

$$\begin{aligned} \# \big \{ (i,j) \in \mathbb {Z}^2 \ : \ Q_{\lambda _n}(i,j) \subset \Omega , \ |(\theta ^{\mathrm {hor}}_n)^{i,j}|> t \text { or } |(\theta ^{\mathrm {ver}}_n)^{i,j}| > t \big \} \leqq C(t) \frac{\delta _n^{3/2}}{\lambda _n}. \end{aligned}$$

Proof

We may assume that $t < \pi$ since otherwise the statement is trivial. Then, if $|(\theta ^{\mathrm {hor}}_n)^{i,j}| > t$ or $|(\theta ^{\mathrm {ver}}_n)^{i,j}| > t$, we get that $\max \{ |\chi _{1,n}^{i,j}|^2, |\chi _{2,n}^{i,j}|^2 \} \geqq \frac{4}{\delta _n} \sin \big ( \frac{t}{2} \big )^2 \geqq C \frac{t^2}{\delta _n}$. Hence, for $\delta _n$ sufficiently small, this implies that $W^\mathrm {d}(\chi _n)^{i,j} \geqq C \frac{t^4}{\delta _n^2}$. Thus we get that

$$\begin{aligned} \begin{aligned} C&\geqq \int _{\Omega } \frac{1}{\varepsilon _n} W^\mathrm {d}(\chi _n) \, {\mathrm {d}{x}} \\&\geqq \lambda _n^2 \# \big \{ (i,j) \in \mathbb {Z}^2 \ : \ Q_{\lambda _n}(i,j) \subset \Omega , \ |(\theta ^{\mathrm {hor}}_n)^{i,j}|> t \text { or } |(\theta ^{\mathrm {ver}}_n)^{i,j}| > t \big \} C \frac{1}{\varepsilon _n}\frac{t^4}{\delta _n^2}. \end{aligned} \end{aligned}$$

Since $\varepsilon _n= \frac{\lambda _n}{\sqrt{\delta _n}}$, this implies the claim. $\square$

A first consequence of the counting argument in Lemma 2.3 is the following estimate on the discrete curl of sequences $\chi _n$ with equibounded energies. For the precise statement, it is convenient to introduce the auxiliary variable ${\overline{\chi }}_n$ defined by

$$\begin{aligned} {\overline{\chi }}_n^{i,j} := \big (\overline{\chi }_{1,n}^{i,j} , \overline{\chi }_{2,n}^{i,j} \big ), \quad \overline{\chi }_{1,n}^{i,j} := \frac{1}{\sqrt{\delta _n}} (\theta ^{\mathrm {hor}}_n)^{i,j} , \quad \overline{\chi }_{2,n}^{i,j} := \frac{1}{\sqrt{\delta _n}} (\theta ^{\mathrm {ver}}_n)^{i,j} . \end{aligned}$$

(2.19)

This is the linearized version of the order parameter $\chi _n$, cf. its definition in (2.8).

Lemma 2.4

Assume that $\sup _n H_n(\chi _n,\Omega ) < + \infty$. Then for every $\Omega ' \subset \subset \Omega$ there exists $C \in (0,+\infty )$ such that

$$\begin{aligned} \Vert \mathrm {curl}^{\mathrm {d}}({{\overline{\chi }}}_n)\Vert _{L^1(\Omega ')} \leqq C \delta _n. \end{aligned}$$

Proof

Let $u_n$ be such that $\chi _n = \chi (u_n)$ as in (2.8). We start by observing that

$$\begin{aligned} \lambda _n\sqrt{\delta _n}\mathrm {curl}^{\mathrm {d}}(\overline{\chi }_n)^{i,j} = (\theta ^{\mathrm {hor}}_n)^{i,j} + (\theta ^{\mathrm {ver}}_n)^{i+1,j} - (\theta ^{\mathrm {ver}}_n)^{i,j} - (\theta ^{\mathrm {hor}}_n)^{i,j+1} \in 2\pi \mathbb {Z}\end{aligned}$$

since $(\theta ^{\mathrm {hor}}_n)^{i,j} + (\theta ^{\mathrm {ver}}_n)^{i+1,j}$ and $(\theta ^{\mathrm {ver}}_n)^{i,j} + (\theta ^{\mathrm {hor}}_n)^{i,j+1}$ both represent an oriented angle between the spins $u_n^{i,j}$ and $u_n^{i+1,j+1}$ and thus must be equal modulo $2 \pi$. Moreover, since $(\theta ^{\mathrm {hor}}_n)^{i,j}, (\theta ^{\mathrm {ver}}_n)^{i+1,j}, (\theta ^{\mathrm {ver}}_n)^{i,j}, (\theta ^{\mathrm {hor}}_n)^{i,j+1} \in [- \pi , \pi )$, we actually get that $\lambda _n\sqrt{\delta _n} \mathrm {curl}^{\mathrm {d}}(\overline{\chi }_n)^{i,j} \in \{-2\pi , 0, 2\pi \}$. If, moreover

$$\begin{aligned} |(\theta ^{\mathrm {hor}}_n)^{i,j}|, \ |(\theta ^{\mathrm {ver}}_n)^{i+1,j}|, \ |(\theta ^{\mathrm {ver}}_n)^{i,j}|, \ |(\theta ^{\mathrm {hor}}_n)^{i,j+1}| < \frac{\pi }{2}, \end{aligned}$$

(2.20)

then we even have that $\lambda _n\sqrt{\delta _n} \mathrm {curl}^{\mathrm {d}}(\overline{\chi }_n)^{i,j} = 0$. For n large enough all cells $Q_{\lambda _n}(i,j)$ that intersect $\Omega '$ as well as all their neighboring cells are contained in $\Omega$. As a consequence, by Lemma 2.3 we have that (2.20) only fails on a subset of $\Omega '$ of measure less than $\lambda _n^2 C \frac{\delta _n^{3/2}}{\lambda _n}$. Hence we conclude that

$$\begin{aligned} \Vert \mathrm {curl}^{\mathrm {d}}({{\overline{\chi }}}_n)\Vert _{L^1(\Omega ')} \leqq \frac{1}{\lambda _n\sqrt{\delta _n}} \cdot 2 \pi \lambda _n^2 C \frac{\delta _n^{3/2}}{\lambda _n} = C \delta _n. \end{aligned}$$

$\square$

Remark 2.5

Lemma 2.4 implies, in particular, that

$$\begin{aligned} \mathrm {curl}^{\mathrm {d}}(\chi _n) \rightharpoonup 0 \quad \text {in the sense of distributions.} \end{aligned}$$

(2.21)

Indeed, using the inequality $\big |2\sin \big (\frac{s}{2}\big ) - s \big | \leqq \frac{1}{24} |s|^3$ and writing $\chi _n$ in terms of ${\overline{\chi }}_n$, we get $|\chi _n^{i,j} - {{\overline{\chi }}}_n^{i,j}|^2 \leqq C \delta _n^2 |{{\overline{\chi }}}_n^{i,j}|^6 \leqq C \delta _n|\chi _n^{i,j}|^4$, where we have used that $|{{\overline{\chi }}}_n^{i,j}| \leqq C |\chi _n^{i,j}| \leqq \frac{C}{\sqrt{\delta _n}}$. Thus, the $L^4$-bounds on $\chi _n$ obtained in Remark 2.2 yield $\Vert \chi _n - {{\overline{\chi }}}_n \Vert _{L^2(\Omega )} \leqq C \delta _n$. A discrete integration by parts shows that $\mathrm {curl}^{\mathrm {d}}(\chi _n - {{\overline{\chi }}}_n) \rightarrow 0$ in $\mathcal {D}'(\Omega )$ and together with Lemma 2.4 we obtain the claim.

As an alternative to the discrete integration by parts, we can observe that $\mathrm {curl}^{\mathrm {d}}(\chi _n - \overline{\chi }_n) = - \mathrm {div}(\mathcal {I}(\chi _n^\perp - \overline{\chi }_n^\perp ))$, where $\mathcal {I}$ is defined by (2.4). Since for every open $\Omega ' \subset \subset \Omega$ we have that $\Vert \mathcal {I}(\chi _n^\perp - \overline{\chi }_n^\perp )\Vert _{L^2(\Omega ')} \leqq 2 \Vert \chi _n - \overline{\chi }_n \Vert _{L^2(\Omega )}$ for n large enough, this allows us to assert that in fact $\mathrm {curl}^{\mathrm {d}}(\chi _n - \overline{\chi }_n) \rightarrow 0$ strongly in $H^{-1}$ locally in $\Omega$. We will later make use of this observation (cf. Proposition 5.2, Step 5).

As we have observed previously, Remark 2.5 suggests that the functionals $H_n$ share similarities with the Aviles–Giga functionals. To give a rigorous statement, we introduce the auxiliary functionals $H_n^*$ defined as follows: for $\chi \in L^1_{\mathrm {loc}}(\mathbb {R}^2;\mathbb {R}^2)$, we set

$$\begin{aligned} H_n^*(\chi ,\Omega ) := \frac{1}{2} \int _\Omega \frac{1}{\varepsilon _n} W(\chi ) + \varepsilon _n|\mathrm {D}^{\mathrm {d}}\chi |^2 \, {\mathrm {d}{x}} \end{aligned}$$

(2.22)

if $\chi = \chi (u)$ as in (2.8) for some $u \in \mathcal {PC}_{\lambda _n}(\mathbb {S}^1)$, and $H_n^*$ extended to $+\infty$ otherwise, where W is defined by (2.18). Up to replacing the condition $\mathrm {curl}^{\mathrm {d}}(\chi _n) \rightharpoonup 0$ (cf. Remark 2.5) with the condition $\mathrm {curl}^{\mathrm {d}}(\chi _n) \equiv 0$, the functionals $H_n^*$ are the discrete Aviles–Giga energies $AG^{\mathrm {d}}_n:L^1_{\mathrm {loc}}(\mathbb {R}^2;\mathbb {R}) {\times }\mathcal {A}\rightarrow [0,+\infty ]$ defined by

$$\begin{aligned} AG^{\mathrm {d}}_n(\varphi ,\Omega ) := \frac{1}{2} \int _\Omega \frac{1}{\varepsilon _n} W(\mathrm {D}^{\mathrm {d}}\varphi ) + \varepsilon _n|\mathrm {D}^{\mathrm {d}}\mathrm {D}^{\mathrm {d}}\varphi |^2 \, {\mathrm {d}{x}} \end{aligned}$$

(2.23)

if $\varphi \in \mathcal {PC}_{\lambda _n}(\mathbb {R})$, and $AG^{\mathrm {d}}_n$ extended to $+\infty$ otherwise.² In the next proposition we prove that the energy bound $H_n(\chi ,\Omega ) \leqq C$ implies a local bound on the energies $H_n^*$. Note that the functionals $H_n^*$ feature the full discrete derivative matrix of $\chi$, and not just the discrete divergence-type term $A^\mathrm {d}(\chi )$ as the functionals $H_n$. Nonetheless, for sequences $\chi _n$ with equibounded energies $H_n$, the full discrete derivative matrix can be controlled by exploiting the vanishing curl condition obtained in Lemma 2.4. Our proof of this fact is inspired by the well-known technique used to prove $H^2$-regularity for weak solutions of elliptic second order PDE.

Proposition 2.6

Let $(\chi _n)_n \in L^1_{\mathrm {loc}}(\mathbb {R}^2;\mathbb {R}^2)$. We have that

$$\begin{aligned} \sup _n H_n(\chi _n,\Omega )< +\infty \quad \implies \quad \sup _n H_n^*(\chi _n,\Omega ') < +\infty \quad \text {for every } \Omega ' \subset \subset \Omega . \end{aligned}$$

Proof

Step 1. (Bound on the derivative term in $H_n^*$.) We claim that

$$\begin{aligned} \sup _n \int _{\Omega '} \varepsilon _n|\mathrm {D}^{\mathrm {d}}{{\overline{\chi }}}_n|^2\, {\mathrm {d}{x}} < +\infty \quad \text {for every } \Omega ' \subset \subset \Omega . \end{aligned}$$

(2.24)

Note that the 1-Lipschitz continuity of the map $s \mapsto \frac{2}{\sqrt{\delta _n}}\sin \big (\frac{\sqrt{\delta _n}}{2} s \big )$ and the definition of $\chi$ in (2.8) and of ${\overline{\chi }}$ in (2.19) imply that $|\mathrm {D}^{\mathrm {d}}\chi _n| \leqq |\mathrm {D}^{\mathrm {d}}{{\overline{\chi }}}_n|$ and thus

$$\begin{aligned} \sup _n \int _{\Omega '} \varepsilon _n|\mathrm {D}^{\mathrm {d}}\chi _n|^2\, {\mathrm {d}{x}} < +\infty \quad \text {for every } \Omega ' \subset \subset \Omega , \end{aligned}$$

(2.25)

providing the first bound needed for $H_n^*$. For later use let us note that using (2.8), (2.19), and the 1-Lipschitz continuity of the map $s \mapsto \frac{1}{\sqrt{\delta _n}}\sin \big (\sqrt{\delta _n} s \big )$ we also get that $|\mathrm {D}^{\mathrm {d}}{\widetilde{\chi }}_n| \leqq |\mathrm {D}^{\mathrm {d}}\overline{\chi }_n|$ and, as a consequence,

$$\begin{aligned}&\sup _n \int _{\Omega '} \varepsilon _n|\mathrm {D}^{\mathrm {d}}{\widetilde{\chi }}_n|^2\, {\mathrm {d}{x}} \nonumber \\&\quad < +\infty \quad \text {for every } \Omega ' \subset \subset \Omega . \end{aligned}$$

(2.26)

To prove (2.24) let us start by considering an additional open set $\Omega ''$ with $\Omega ' \subset \subset \Omega '' \subset \subset \Omega$ and a smooth cut-off function $\zeta \in C_c^\infty (\Omega '';[0,1])$ with $\zeta \equiv 1$ on a neighborhood of $\overline{ \Omega '}$. Although not necessary, it will be convenient for our computations to introduce the discretizations $\zeta _n \in \mathcal {PC}_{\lambda _n}([0,1])$ by $\zeta _n^{i,j} := \zeta (\lambda _n(i,j))$. Next, let us observe that by (2.8) and (2.19) we have that

$$\begin{aligned} A^\mathrm {d}(\chi _n)^{i,j} = \partial ^{\mathrm {d}}_1 \bigg ( \frac{1}{\sqrt{\delta _n}} \sin (\sqrt{\delta _n} \overline{\chi }_{1,n}) \bigg )^{i-1,j} + \partial ^{\mathrm {d}}_2 \bigg ( \frac{1}{\sqrt{\delta _n}} \sin (\sqrt{\delta _n} \overline{\chi }_{2,n}) \bigg )^{i,j-1} \, . \end{aligned}$$

Therefore, using twice a discrete integration by parts, we get that

$$\begin{aligned} \begin{aligned} I_n&:= \int _{\mathbb {R}^2} A^\mathrm {d}(\chi _n)\partial ^{\mathrm {d}}_1 (|\zeta _n|^2 \overline{\chi }_{1,n})^{{\varvec{\cdot }}- e_1} \, {\mathrm {d}{x}} \\&= \int _{\mathbb {R}^2} \partial ^{\mathrm {d}}_1 \bigg ( \frac{1}{\sqrt{\delta _n}} \sin (\sqrt{\delta _n} \overline{\chi }_{1,n}) \bigg ) \partial ^{\mathrm {d}}_1 (|\zeta _n|^2 \overline{\chi }_{1,n}) \\&+ \partial ^{\mathrm {d}}_1 \bigg ( \frac{1}{\sqrt{\delta _n}} \sin (\sqrt{\delta _n} \overline{\chi }_{2,n}) \bigg ) \partial ^{\mathrm {d}}_2 (|\zeta _n|^2 \overline{\chi }_{1,n}) \, {\mathrm {d}{x}}. \end{aligned} \end{aligned}$$

(2.27)

In what follows, we show how (2.27) can be used to deduce the bound $\int _{\Omega '} \varepsilon _n|\mathrm {D}^{\mathrm {d}}\overline{\chi }_{1,n}|^2 \, {\mathrm {d}{x}} \leqq C$. The remaining bound $\int _{\Omega '} \varepsilon _n|\mathrm {D}^{\mathrm {d}}\overline{\chi }_{2,n}|^2 \, {\mathrm {d}{x}} \leqq C$ can be proved analogously starting instead from the equation

$$\begin{aligned} \begin{aligned}&\int _{\mathbb {R}^2} A^\mathrm {d}(\chi _n) \, \partial ^{\mathrm {d}}_2 (|\zeta _n|^2 \overline{\chi }_{2,n})^{{\varvec{\cdot }}- e_2} \, {\mathrm {d}{x}} \\&\quad = \int _{\mathbb {R}^2} \partial ^{\mathrm {d}}_2 \bigg ( \frac{1}{\sqrt{\delta _n}} \sin (\sqrt{\delta _n} \overline{\chi }_{1,n}) \bigg ) \partial ^{\mathrm {d}}_1 (|\zeta _n|^2 \overline{\chi }_{2,n}) \\&\qquad + \partial ^{\mathrm {d}}_2 \bigg ( \frac{1}{\sqrt{\delta _n}} \sin (\sqrt{\delta _n} \overline{\chi }_{2,n}) \bigg ) \partial ^{\mathrm {d}}_2 (|\zeta _n|^2 \overline{\chi }_{2,n}) \, {\mathrm {d}{x}}. \end{aligned} \end{aligned}$$

We rewrite the right-hand side of (2.27) by using a discrete chain rule and a particular version of a discrete product rule which takes the form $\partial ^{\mathrm {d}}_k (v w) = \frac{1}{2} (w + w^{{\varvec{\cdot }}+ e_k}) \partial ^{\mathrm {d}}_k v + \frac{1}{2} (v + v^{{\varvec{\cdot }}+ e_k}) \partial ^{\mathrm {d}}_k w$ for $v, w \in \mathcal {PC}_{\lambda _n}(\mathbb {R})$. We obtain that

$$\begin{aligned} \begin{aligned} I_n&= \frac{1}{2} \int _{\mathbb {R}^2} \big ( |\zeta _n|^2 + |\zeta _n^{{\varvec{\cdot }}+ e_1}|^2 \big ) \cos (\sqrt{\delta _n} X_{1,n}) |\partial ^{\mathrm {d}}_1 \overline{\chi }_{1,n}|^2 \\&\qquad + \big ( |\zeta _n|^2 + |\zeta _n^{{\varvec{\cdot }}+ e_2}|^2 \big ) \cos (\sqrt{\delta _n} X_{2,n}) |\partial ^{\mathrm {d}}_2 \overline{\chi }_{1,n}|^2 \, {\mathrm {d}{x}} + R_n, \end{aligned} \end{aligned}$$

(2.28)

where

$$\begin{aligned} \begin{aligned} R_n&= \frac{1}{2} \int _{\mathbb {R}^2} \cos (\sqrt{\delta _n} X_{1,n}) \partial ^{\mathrm {d}}_1 \overline{\chi }_{1,n} \big (\overline{\chi }_{1,n} + \overline{\chi }_{1,n}^{{\varvec{\cdot }}+ e_1} \big ) \partial ^{\mathrm {d}}_1 (|\zeta _n|^2) \\&\qquad + \cos (\sqrt{\delta _n} X_{2,n}) \partial ^{\mathrm {d}}_1 \overline{\chi }_{2,n} \big (\overline{\chi }_{1,n} + \overline{\chi }_{1,n}^{{\varvec{\cdot }}+ e_2} \big ) \partial ^{\mathrm {d}}_2 (|\zeta _n|^2) \\&\qquad + \big ( |\zeta _n|^2 + |\zeta _n^{{\varvec{\cdot }}+ e_2}|^2 \big ) \cos (\sqrt{\delta _n} X_{2,n}) \partial ^{\mathrm {d}}_2 \overline{\chi }_{1,n} (\partial ^{\mathrm {d}}_1 \overline{\chi }_{2,n} - \partial ^{\mathrm {d}}_2 \overline{\chi }_{1,n}) \, {\mathrm {d}{x}}, \end{aligned} \end{aligned}$$

and where $X_{1,n}^{i,j}$ is an intermediate point between $\overline{\chi }_{1,n}^{i,j}$ and $\overline{\chi }_{1,n}^{i+1,j}$ and $X_{2,n}^{i,j}$ lies between $\overline{\chi }_{2,n}^{i,j}$ and $\overline{\chi }_{2,n}^{i+1,j}$. In the following, we may restrict all integrations to $\Omega ''$ with the understanding that the resulting estimates hold for n large enough. To estimate $R_n$ let us recall that by Lemma 2.4 we have that $\Vert \partial ^{\mathrm {d}}_1 \overline{\chi }_{2,n} - \partial ^{\mathrm {d}}_2 \overline{\chi }_{1,n} \Vert _{L^1(\Omega '')} = \Vert \mathrm {curl}^{\mathrm {d}}(\overline{\chi }_n) \Vert _{L^1(\Omega '')} \leqq C \delta _n$. Moreover, $|\partial ^{\mathrm {d}}_2 \overline{\chi }_{1,n}| \leqq \frac{C}{\lambda _n\sqrt{\delta _n}}$ and, as a consequence,

$$\begin{aligned}&\int _{\mathbb {R}^2} \big | \big ( |\zeta _n|^2 + |\zeta _n^{{\varvec{\cdot }}+ e_2}|^2 \big ) \cos (\sqrt{\delta _n} X_{2,n}) \partial ^{\mathrm {d}}_2 \overline{\chi }_{1,n} (\partial ^{\mathrm {d}}_1 \overline{\chi }_{2,n} - \partial ^{\mathrm {d}}_2 \overline{\chi }_{1,n}) \big | \, {\mathrm {d}{x}} \nonumber \\&\quad \leqq C \frac{\sqrt{\delta _n}}{\lambda _n} = \frac{C}{\varepsilon _n}. \end{aligned}$$

(2.29)

Furthermore, we have that $\partial ^{\mathrm {d}}_k (|\zeta _n|^2) = \frac{1}{\lambda _n} |\zeta _n^{{\varvec{\cdot }}+ e_k} - \zeta _n| \, | \zeta _n^{{\varvec{\cdot }}+ e_k} + \zeta _n| \leqq C ( \zeta _n^{{\varvec{\cdot }}+ e_k} + \zeta _n)$ because $\mathrm {D}^{\mathrm {d}}\zeta _n$ are bounded in $L^\infty (\mathbb {R}^2)$. Using Young’s inequality we get that

$$\begin{aligned} \begin{aligned}&\int _{\mathbb {R}^2} \big | \cos (\sqrt{\delta _n} X_{1,n}) \partial ^{\mathrm {d}}_1 \overline{\chi }_{1,n} \big (\overline{\chi }_{1,n} + \overline{\chi }_{1,n}^{{\varvec{\cdot }}+ e_1} \big ) \partial ^{\mathrm {d}}_1 (|\zeta _n|^2) \big | \, {\mathrm {d}{x}} \\&\qquad \leqq C \int _{\Omega ''} \frac{1}{M} |\partial ^{\mathrm {d}}_1 \overline{\chi }_{1,n}|^2 \big (|\zeta _n|^2 + |\zeta _n^{{\varvec{\cdot }}+ e_1}|^2 \big ) + M |\overline{\chi }_{1,n}|^2 \, {\mathrm {d}{x}}, \end{aligned} \end{aligned}$$

(2.30)

where M is an arbitrary positive number. Similarly, using first the triangle inequality, we also get the estimate

$$\begin{aligned} \begin{aligned}&\int _{\mathbb {R}^2} \big | \cos (\sqrt{\delta _n} X_{2,n}) \partial ^{\mathrm {d}}_1 \overline{\chi }_{2,n} \big (\overline{\chi }_{1,n} + \overline{\chi }_{1,n}^{{\varvec{\cdot }}+ e_2} \big ) \partial ^{\mathrm {d}}_2 (|\zeta _n|^2) \big | \, {\mathrm {d}{x}} \\&\leqq \int _{\mathbb {R}^2} \big | \partial ^{\mathrm {d}}_2 \overline{\chi }_{1,n} \big (\overline{\chi }_{1,n} + \overline{\chi }_{1,n}^{{\varvec{\cdot }}+ e_2} \big ) \partial ^{\mathrm {d}}_2 (|\zeta _n|^2) \big | \\&\quad + \big | \mathrm {curl}^{\mathrm {d}}(\overline{\chi }_n) \big (\overline{\chi }_{1,n} + \overline{\chi }_{1,n}^{{\varvec{\cdot }}+ e_2} \big ) \partial ^{\mathrm {d}}_2 (|\zeta _n|^2) \big | \, {\mathrm {d}{x}} \\&\leqq C \int _{\Omega ''} \frac{1}{M} |\partial ^{\mathrm {d}}_2 \overline{\chi }_{1,n}|^2 \big (|\zeta _n|^2 + |\zeta _n^{{\varvec{\cdot }}+ e_2}|^2 \big ) + M |\overline{\chi }_{1,n}|^2 \, {\mathrm {d}{x}} + C \sqrt{\delta _n}, \end{aligned} \end{aligned}$$

(2.31)

where we have used Lemma 2.4 and the fact that $|\overline{\chi }_{1,n}| \leqq \frac{C}{\sqrt{\delta _n}}$. By (2.29)–(2.31) we get that

$$\begin{aligned} |R_n|\leqq & {} \frac{C}{M} \int _{\Omega ''} |\partial ^{\mathrm {d}}_1 \overline{\chi }_{1,n}|^2 \big (|\zeta _n|^2 + |\zeta _n^{{\varvec{\cdot }}+ e_1}|^2 \big ) + |\partial ^{\mathrm {d}}_2 \overline{\chi }_{1,n}|^2 \big (|\zeta _n|^2 + |\zeta _n^{{\varvec{\cdot }}+ e_2}|^2 \big ) \, {\mathrm {d}{x}} \\&+ CM + \frac{C}{\varepsilon _n}, \end{aligned}$$

where we have used that $\sqrt{\delta _n} \leqq \frac{1}{\varepsilon _n}$ for n large enough (by (2.13)) and the fact that $\overline{\chi }_{1,n}$ are bounded in $L^2(\Omega '')$. The latter bound is due to Remark 2.2 and the fact that $|\overline{\chi }_n| \leqq \frac{\pi }{2} |\chi _n|$. With the bound on $R_n$ in place, we now return to (2.28) and estimate $I_n$ from below as follows:

$$\begin{aligned} \begin{aligned} I_n&\geqq \Big ( \frac{1}{4} - \frac{C}{M} \Big ) \int _{\Omega ''} |\partial ^{\mathrm {d}}_1 \overline{\chi }_{1,n}|^2 \big (|\zeta _n|^2 + |\zeta _n^{{\varvec{\cdot }}+ e_1}|^2 \big ) + |\partial ^{\mathrm {d}}_2 \overline{\chi }_{1,n}|^2 \big (|\zeta _n|^2 + |\zeta _n^{{\varvec{\cdot }}+ e_2}|^2 \big ) \, {\mathrm {d}{x}}\\&\quad + \frac{1}{2} \int _{\Omega ''} |\partial ^{\mathrm {d}}_1 \overline{\chi }_{1,n}|^2 \big (|\zeta _n|^2 + |\zeta _n^{{\varvec{\cdot }}+ e_1}|^2 \big ) \big ( \cos (\sqrt{\delta _n} X_{1,n}) - \tfrac{1}{2} \big ) \\&\quad + |\partial ^{\mathrm {d}}_2 \overline{\chi }_{1,n}|^2 \big (|\zeta _n|^2 + |\zeta _n^{{\varvec{\cdot }}+ e_2}|^2 \big ) \big ( \cos (\sqrt{\delta _n} X_{2,n}) - \tfrac{1}{2} \big ) \, {\mathrm {d}{x}} \\&\quad - CM - \frac{C}{\varepsilon _n}. \end{aligned} \end{aligned}$$

(2.32)

For all indices $(i,j) \in \mathbb {Z}^2$ such that

$$\begin{aligned} |\overline{\chi }_{1,n}^{i,j}|, \ |\overline{\chi }_{1,n}^{i+1,j}|, \ |\overline{\chi }_{2,n}^{i,j}|, \ |\overline{\chi }_{2,n}^{i+1,j}| \leqq \frac{\arccos {\frac{1}{2}}}{\sqrt{\delta _n}}, \end{aligned}$$

(2.33)

we have that $\cos (\sqrt{\delta _n} X_{1,n}), \cos (\sqrt{\delta _n} X_{2,n}) \geqq \frac{1}{2}$ on the cell $Q_{\lambda _n}(i,j)$. On the other hand, for n large enough all cells $Q_{\lambda _n}(i,j)$ that intersect $\Omega ''$ as well as all their neighboring cells are contained in $\Omega$ and thus in view of (2.19), Lemma 2.3 implies that

$$\begin{aligned} \# \big \{ (i,j) \in \mathbb {Z}^2 \ : \ Q_{\lambda _n}(i,j) \cap \Omega '' \ne \emptyset \,\text {and}\,{2.33}\,\text {fails} \big \} \leqq C \frac{\delta _n^{3/2}}{\lambda _n}. \end{aligned}$$

This allows us to estimate the second integral in (2.32) from below by splitting it into the integral on the cells where (2.33) holds and the integral on the cells where it fails: On the former, the integrand is non-negative. On the latter cells, we use that $|\mathrm {D}^{\mathrm {d}}\overline{\chi }_{1,n}|^2 \leqq \frac{C}{\lambda _n^2 \delta _n}$ and consequently obtain that the second integral in (2.32) is bounded from below by $-C \lambda _n^2 \frac{\delta _n^{3/2}}{\lambda _n} \frac{1}{\lambda _n^2 \delta _n} = - \frac{C}{\varepsilon _n}$. Thus,

$$\begin{aligned} I_n&\geqq \Big ( \frac{1}{4} - \frac{C}{M} \Big ) \int _{\Omega ''} |\partial ^{\mathrm {d}}_1 \overline{\chi }_{1,n}|^2 \big (|\zeta _n|^2 + |\zeta _n^{{\varvec{\cdot }}+ e_1}|^2 \big ) \nonumber \\&\quad + |\partial ^{\mathrm {d}}_2 \overline{\chi }_{1,n}|^2 \big (|\zeta _n|^2 + |\zeta _n^{{\varvec{\cdot }}+ e_2}|^2 \big ) \, {\mathrm {d}{x}} - CM - \frac{C}{\varepsilon _n}. \end{aligned}$$

(2.34)

To find the desired $L^2$ estimate on $\mathrm {D}^{\mathrm {d}}\overline{\chi }_{1,n}$, we combine this lower bound with an upper bound on the left-hand side of (2.27). Using Young’s inequality, a discrete product rule, and the bound on the energy $H_n$ we get that

$$\begin{aligned} \begin{aligned} I_n&\leqq \frac{1}{2} \int _{\Omega ''} M A^\mathrm {d}(\chi _n)^2 + \frac{1}{M} \big | \partial ^{\mathrm {d}}_1 (|\zeta _n|^2 \overline{\chi }_{1,n}) \big |^2 \, {\mathrm {d}{x}} \\&\leqq \frac{CM}{\varepsilon _n} + \frac{1}{M} \int _{\Omega ''} \big |\partial ^{\mathrm {d}}_1 (|\zeta _n|^2) \big |^2 |\overline{\chi }_{1,n}^{{\varvec{\cdot }}+ e_1}|^2 + |\zeta _n|^4 |\partial ^{\mathrm {d}}_1 \overline{\chi }_{1,n}|^2 \, {\mathrm {d}{x}}. \end{aligned} \end{aligned}$$

Finally, as already observed in this proof, we use that $\overline{\chi }_{1,n}$ are bounded in $L^2(\Omega )$, $\mathrm {D}^{\mathrm {d}}(|\zeta _n|^2)$ are bounded in $L^\infty$, and $|\zeta _n|^4 \leqq |\zeta _n|^2 \leqq |\zeta _n|^2 + |\zeta _n^{{\varvec{\cdot }}+ e_1}|^2$ to obtain that

$$\begin{aligned} I_n \leqq \frac{CM}{\varepsilon _n} + \frac{C}{M} + \frac{1}{M} \int _{\Omega ''} \big ( |\zeta _n|^2 + |\zeta _n^{{\varvec{\cdot }}+ e_1}|^2 \big ) |\partial ^{\mathrm {d}}_1 \overline{\chi }_{1,n}|^2 \, {\mathrm {d}{x}}. \end{aligned}$$

Together with (2.34) this implies that

$$\begin{aligned}&\Big ( \frac{1}{4} - \frac{C}{M} \Big ) \int _{\Omega ''} |\partial ^{\mathrm {d}}_1 \overline{\chi }_{1,n}|^2 \big (|\zeta _n|^2 + |\zeta _n^{{\varvec{\cdot }}+ e_1}|^2 \big ) + |\partial ^{\mathrm {d}}_2 \overline{\chi }_{1,n}|^2 \big (|\zeta _n|^2 + |\zeta _n^{{\varvec{\cdot }}+ e_2}|^2 \big ) \, {\mathrm {d}{x}} \\&\quad \leqq \frac{C}{\varepsilon _n} (1 + M) + \frac{C}{M}. \end{aligned}$$

As none of the constants C depend on M, choosing M sufficiently large and if n is large enough, the left-hand side provides an upper bound on $\frac{1}{8} \int _{\Omega '} |\mathrm {D}^{\mathrm {d}}\overline{\chi }_{1,n}|^2 \, {\mathrm {d}{x}}$. Thus we get that $\int _{\Omega '} \varepsilon _n|\mathrm {D}^{\mathrm {d}}\overline{\chi }_{1,n}|^2 \, {\mathrm {d}{x}} \leqq C$ as desired.

Step 2. (Bound on the potential term in $H_n^*$.) We claim that

$$\begin{aligned} \sup _n \int _{\Omega '} \frac{1}{\varepsilon _n} W(\chi _n) \, \, {\mathrm {d}{x}} < +\infty \quad \text {for every } \Omega ' \subset \subset \Omega . \end{aligned}$$

(2.35)

By the reverse triangle inequality we have that

$$\begin{aligned} \begin{aligned}&\big | \sqrt{W^\mathrm {d}}(\chi _n^{i,j}) - \sqrt{W}(\chi _n^{i,j}) \big | \\&\leqq \frac{1}{2} \Big | 2 - |\chi _{1,n}^{i,j}|^2 - |\chi _{1,n}^{i-1,j}|^2 - |\chi _{2,n}^{i,j}|^2 - |\chi _{2,n}^{i,j-1}|^2 \\&\quad - \big ( 2 - 2|\chi _{1,n}^{i,j}|^2 - 2|\chi _{2,n}^{i,j}|^2 \big ) \Big | \\&= \frac{1}{2} \big | (\chi _{1,n}^{i,j} + \chi _{1,n}^{i-1,j}) \lambda _n\partial ^{\mathrm {d}}_1 \chi _{1,n}^{i-1,j} + (\chi _{2,n}^{i,j} + \chi _{2,n}^{i,j-1}) \lambda _n\partial ^{\mathrm {d}}_2 \chi _{2,n}^{i,j-1} \big |. \end{aligned} \end{aligned}$$

(2.36)

Let $\Omega ''$ be another open set with $\Omega ' \subset \subset \Omega '' \subset \subset \Omega$. Using (2.25), the fact that $|\chi _{1,n}|,|\chi _{2,n}| \leqq \frac{C}{\sqrt{\delta _n}}$, and (2.13), we obtain for n large enough that

$$\begin{aligned}&\frac{1}{\sqrt{\varepsilon _n}} \big \Vert \sqrt{W^\mathrm {d}}(\chi _n^{i,j}) - \sqrt{W}(\chi _n^{i,j}) \big \Vert _{L^2(\Omega ')} \leqq C \frac{\lambda _n}{\sqrt{\varepsilon _n} \sqrt{\delta _n}} \Vert \mathrm {D}^{\mathrm {d}}\chi \Vert _{L^2(\Omega '')}\\&\quad \leqq C \frac{\lambda _n}{\varepsilon _n\sqrt{\delta _n}} = C. \end{aligned}$$

Writing $W^\mathrm {d}- W = \big ( 2 \sqrt{W^\mathrm {d}} - (\sqrt{W^\mathrm {d}} - \sqrt{W}) \big ) \big ( \sqrt{W^\mathrm {d}} - \sqrt{W} \big )$, we infer that

$$\begin{aligned} \int _{\Omega '} \frac{1}{\varepsilon _n} \big | W^\mathrm {d}(\chi _n) - W(\chi _n) \big | \, {\mathrm {d}{x}} \leqq \Big ( \tfrac{2}{\sqrt{\varepsilon _n}} \big \Vert \sqrt{W^\mathrm {d}}(\chi _n) \big \Vert _{L^2(\Omega ')} + C \Big ) \cdot C \leqq C, \end{aligned}$$

where we have used that $H_n(\chi _n, \Omega ) \leqq C$ implies that $\big \Vert \sqrt{W^\mathrm {d}}(\chi _n) \big \Vert _{L^2(\Omega ')} \leqq C \sqrt{\varepsilon _n}$.

This concludes the proof. $\square$

We conclude the section by investigating a first consequence of Proposition 2.6. For the classical Aviles–Giga functionals in dimension two it is known that a uniform bound on the energies $AG_{\varepsilon }(\varphi _{\varepsilon },\Omega )$ implies a bound on $\nabla \varphi _{\varepsilon }$ not only in $L^4(\Omega )$ but even in $L^6(\Omega )$ (cf. [5, Theorem 6.1]). Using Proposition 2.6 and exploiting the analogy between $H_n^*$ and the classical Aviles–Giga, in the next proposition we improve the $L^4$ bound obtained in Remark 2.2.

Proposition 2.7

Let $(\chi _n)_n \in L^1_{\mathrm {loc}}(\mathbb {R}^2; \mathbb {R}^2)$ and assume that $\sup _n H_n(\chi _n, \Omega ) < +\infty$. Then, for every $\Omega ' \subset \subset \Omega$, $(\chi _n)_n$ is bounded in $L^6(\Omega ')$.

Proof

We let $\Omega ' \subset \subset \Omega$ be fixed. We start by introducing a piecewise affine interpolation ${\widehat{\chi }}_n$ of the discrete functions $|\chi _n|$. To this end, let $T_{\lambda _n}^-(i,j)$ and $T_{\lambda _n}^+(i,j)$ be the two triangles partitioning the cell $Q_{\lambda _n}(i,j)$ defined by

$$\begin{aligned} T_{\lambda _n}^-&:= \{ \lambda _n(i, j) + \lambda _ny \in Q_{\lambda _n}(i,j) \ : \ y_1 \in [0,1], y_2 \in [0,1-y_1] \}, \\ T_{\lambda _n}^+&:= \{ \lambda _n(i, j) + \lambda _ny \in Q_{\lambda _n}(i,j) \ : \ y_1 \in (0,1) , y_2 \in (1-y_1, 1) \}. \end{aligned}$$

We define the function ${\widehat{\chi }}_n$ on $T_{\lambda _n}^-(i,j)$ by interpolating the values on the three vertices of $T_{\lambda _n}^-(i,j)$, i.e.,

$$\begin{aligned} {\widehat{\chi }}_n (\lambda _n(i,j) + \lambda _ny) := (1 - y_1 - y_2) |\chi _n^{i,j}| + y_1 |\chi _n^{i+1,j}| + y_2 |\chi _n^{i,j+1}|. \end{aligned}$$

Analogously, for $\lambda _n(i,j) + \lambda _ny \in T_{\lambda _n}^+(i,j)$,

$$\begin{aligned} {\widehat{\chi }}_n (\lambda _n(i,j) + \lambda _ny):= & {} (1 - y_1) |\chi _n^{i,j+1}| + (1 - y_2) |\chi _n^{i+1,j}| \\&+ (y_1 + y_2 - 1) |\chi _n^{i+1,j+1}|. \end{aligned}$$

Below, we will exploit Sobolev embeddings to show that $({\widehat{\chi }}_n)_n$ is bounded in $L^6(\Omega '')$ for some open set $\Omega ''$ with $\Omega ' \subset \subset \Omega '' \subset \subset \Omega$. This will conclude the proof since we can control the $L^6$ norm of $\chi _n$ by that of ${\widehat{\chi }}_n$ as follows: Given any $(i,j) \in \mathbb {Z}^2$, on the sub-triangle

$$\begin{aligned}&T^{1/2}_{\lambda _n}(i,j) := \{ \lambda _n(i, j) + \lambda _ny \in Q_{\lambda _n}(i,j) :\\&\quad y_1 \in [0,\tfrac{1}{2}], y_2 \in [0, \tfrac{1}{2}-y_1] \} \subset T^-_{\lambda _n}(i,j) \end{aligned}$$

we have that $|{\widehat{\chi }}_n| \geqq \frac{1}{2} |\chi _n^{i,j}|$. Therefore,

$$\begin{aligned} \Vert {\widehat{\chi }}_n \Vert _{L^6(Q_{\lambda _n}(i,j))}^6 \geqq C \mathcal {L}^2 \big (T^{1/2}_{\lambda _n}(i,j) \big ) |\chi _n^{i,j}|^6 = C \Vert \chi _n\Vert _{L^6(Q_{\lambda _n(i,j)})}^6, \end{aligned}$$

where we have used that $\mathcal {L}^2 \big (T^{1/2}_{\lambda _n}(i,j) \big ) = C \mathcal {L}^2(Q_{\lambda _n}(i,j))$ with C independent of n, i, j. For all n large enough, every cell $Q_{\lambda _n}(i,j)$ that intersects $\Omega '$ is contained in $\Omega ''$ and thus we conclude that

$$\begin{aligned} \Vert \chi _n\Vert _{L^6(\Omega ')} \leqq C \Vert {\widehat{\chi }}_n \Vert _{L^6(\Omega '')} \end{aligned}$$

(2.37)

for all n large enough.

To estimate ${\widehat{\chi }}_n$ in $L^6$, let us fix an open and smooth set $\Omega ''$ and an additional open set $\Omega '''$ satisfying $\Omega ' \subset \subset \Omega '' \subset \subset \Omega ''' \subset \subset \Omega$. We observe that ${\widehat{\chi }}_n$ belongs to $W^{1, \infty }_{\mathrm {loc}}(\mathbb {R}^2; \mathbb {R})$ with a Sobolev gradient that is constant on $T_{\lambda _n}^{\pm }(i,j)$ and given by

$$\begin{aligned} \nabla {\widehat{\chi }}_n&= (\partial ^{\mathrm {d}}_1 |\chi _n|^{i,j}, \partial ^{\mathrm {d}}_2 |\chi _n|^{i,j}) \quad \text {in } T_{\lambda _n}^-(i,j), \\ \nabla {\widehat{\chi }}_n&= (\partial ^{\mathrm {d}}_1 |\chi _n|^{i, j+1} , \partial ^{\mathrm {d}}_2 |\chi _n|^{i+1,j}) \quad \text {in } T_{\lambda _n}^+(i,j). \end{aligned}$$

This entails the estimate $\Vert \nabla {\widehat{\chi }}_n \Vert _{L^2(\Omega '')} \leqq \Vert \mathrm {D}^{\mathrm {d}}|\chi _n| \Vert _{L^2(\Omega ''')}$ for n large enough. By use of the reverse triangle inequality, $\mathrm {D}^{\mathrm {d}}|\chi _n|$ is bounded by $\mathrm {D}^{\mathrm {d}}\chi _n$ and thus, by Proposition 2.6, we get that

$$\begin{aligned} \varepsilon _n\Vert \nabla {\widehat{\chi }}_n \Vert _{L^2(\Omega '')} \leqq C. \end{aligned}$$

(2.38)

Next, we introduce the convex function

$$\begin{aligned} V :\mathbb {R}\rightarrow \mathbb {R}, \quad V(s) := {\left\{ \begin{array}{ll} 0 &{} \text {if } -1< s < 1, \\ s^2 - 1 &{} \text {if } |s| \geqq 1, \end{array}\right. } \end{aligned}$$

and set $V^2(s) := |V(s)|^2$. $V^2$ is the convex envelope of the double-well potential $s \mapsto (1-s^2)^2$. By convexity of $V^2$ and by the definition of ${\widehat{\chi }}_n$ we have that

$$\begin{aligned}&V^2({\widehat{\chi }}_n( \lambda _n(i,j) + \lambda _ny))\\&\quad \leqq (1 - y_1 - y_2) V^2(|\chi _n^{i,j}|)+ y_1 V^2(|\chi _n^{i+1,j}|) + y_2 V^2(|\chi _n^{i,j+1}|) \end{aligned}$$

for $\lambda _n(i,j) + \lambda _ny \in T^-_{\lambda _n}(i,j)$ and

$$\begin{aligned}&V^2({\widehat{\chi }}_n( \lambda _n(i,j) + \lambda _ny)) \leqq (1 - y_1) V^2(|\chi _n^{i,j+1}|) + (1 - y_2) V^2(|\chi _n^{i+1,j}|) \\&\quad + (y_1 + y_2 - 1) V^2(|\chi _n^{i+1,j+1}|) \end{aligned}$$

for $\lambda _n(i,j) + \lambda _ny \in T^+_{\lambda _n}(i,j)$. Since $V^2(|\chi _n|) \leqq (1 - |\chi _n|^2)^2 = W(\chi _n)$, Proposition 2.6 gives us that

$$\begin{aligned} \int _{\Omega ''} \frac{1}{\varepsilon _n} V^2 ({\widehat{\chi }}_n) \, {\mathrm {d}{x}} \leqq \int _{\Omega '''} \frac{1}{\varepsilon _n} W(\chi _n) \, {\mathrm {d}{x}} \leqq C \end{aligned}$$

for n large enough. Combining this with (2.38), we have obtained the following energy bound on ${\widehat{\chi }}_n$:

$$\begin{aligned} \int _{\Omega ''} \frac{1}{\varepsilon _n} V^2 ({\widehat{\chi }}_n) + \varepsilon _n|\nabla {\widehat{\chi }}_n|^2 \, {\mathrm {d}{x}} \leqq C. \end{aligned}$$

(2.39)

Next we introduce a primitive P of the function V, namely the $C^1$ function

$$\begin{aligned} P :\mathbb {R}\rightarrow \mathbb {R}, \quad P(s):= {\left\{ \begin{array}{ll} \frac{1}{3} s^3 - s - \frac{2}{3} &{} \text {if } s \geqq 1, \\ 0 &{} \text {if } -1< s < 1, \\ \frac{1}{3} s^3 - s + \frac{2}{3} &{} \text {if } s \leqq -1. \end{array}\right. } \end{aligned}$$

The function P has cubic growth, i.e.,

$$\begin{aligned} c_1 |s|^3 + c_2 \leqq |P(s)| \leqq C_1 |s|^3 + C_2 \end{aligned}$$

(2.40)

with constants $c_1, C_1 > 0$ and $c_2, C_2 \in \mathbb {R}$. In particular, $P \circ {\widehat{\chi }}_n$ are bounded in $L^1(\Omega '')$ since ${\widehat{\chi }}_n$ are bounded in $L^4(\Omega '')$, being the piecewise affine interpolations of $|\chi _n|$, which are bounded in $L^4(\Omega )$ by Remark 2.2. Moreover, since P is $C^1$ and locally Lipschitz and ${\widehat{\chi }}_n$ belong to $W^{1, \infty }(\Omega ''; \mathbb {R})$, by the chain rule $P \circ {\widehat{\chi }}_n$ are Sobolev functions as well and $\nabla (P \circ {\widehat{\chi }}_n) = (V \circ {\widehat{\chi }}_n) \, \nabla {\widehat{\chi }}_n$. Using Young’s inequality, we get that

$$\begin{aligned} \Vert \nabla (P \circ {\widehat{\chi }}_n) \Vert _{L^1(\Omega '')} \leqq \frac{1}{2} \int _{\Omega ''} \frac{1}{\varepsilon _n} |V ({\widehat{\chi }}_n)|^2 + \varepsilon _n|\nabla {\widehat{\chi }}_n|^2 \, {\mathrm {d}{x}} \leqq C \end{aligned}$$

by (2.39). Thus, $P \circ {\widehat{\chi }}_n$ are bounded in $W^{1,1}(\Omega '')$ and recalling that we have chosen $\Omega ''$ to be a smooth domain, Poincaré’s inequality leads to a bound on $P \circ {\widehat{\chi }}_n$ in $L^2(\Omega '')$. Finally, (2.40) yields $\Vert {\widehat{\chi }}_n \Vert _{L^6(\Omega '')}^6 \leqq \Vert c_1^{-1} (P \circ {\widehat{\chi }}_n - c_2)\Vert _{L^2(\Omega '')}^2 \leqq C$ and thereby the desired $L^6$ bound. Then by (2.37) we conclude the proof. $\square$

3 Entropies and the Limit Functional

In this section we define the notion of entropy that we will use in this paper and define the limit functional H for our energies $H_n$.

Definition 3.1

We say that a map $\Phi :\mathbb {R}^2 \rightarrow \mathbb {R}^2$ is an entropy if $\Phi \in C^\infty _c(\mathbb {R}^2 \setminus \{0\}; \mathbb {R}^2)$ and it satisfies

$$\begin{aligned} \xi \cdot (\mathrm {D}\Phi (\xi ) \xi ^\perp ) = 0 \quad \text {for all } \xi \in \mathbb {R}^2. \end{aligned}$$

(3.1)

We define the space $\mathrm {Ent}:= \{ \Phi \in C^\infty _c(\mathbb {R}^2 \setminus \{0\}; \mathbb {R}^2), \ \Phi \text { is an entropy}\}$.

This notion of entropy strongly resembles the one used in [26]. (There it is not required that $\Phi$ is zero in a neighborhood of zero.) As in [26, Lemma 2.2] we associate to every $\Phi \in \mathrm {Ent}$ a pair of functions $(\Psi ,\alpha )$ defined by

$$\begin{aligned} \alpha (\xi )&:= \frac{\xi ^\perp \cdot (\mathrm {D}\Phi (\xi ) \xi ^\perp )}{|\xi |^2}, \end{aligned}$$

(3.2)

$$\begin{aligned} \Psi (\xi )&:= - \frac{1}{2 |\xi |^2} \big ( \mathrm {D}\Phi (\xi ) - \alpha (\xi ) \mathrm {Id} \big ) \xi . \end{aligned}$$

(3.3)

Note that $\mathrm {supp}(\Psi )$, $\mathrm {supp}(\alpha ) \subset \mathrm {supp}(\Phi ) \subset \mathbb {R}^2 \setminus \{0\}$ and $\Psi \in C_c^\infty (\mathbb {R}^2 \setminus \{0\}; \mathbb {R}^2)$ and $\alpha \in C_c^\infty (\mathbb {R}^2 \setminus \{0\})$, since $\Phi \in C^\infty _c(\mathbb {R}^2 \setminus \{0\}; \mathbb {R}^2)$. This will be useful for technical reasons in the proofs.

Using property (3.1) and the identity $\mathrm {Id} = \frac{1}{|\xi |^2}\xi \otimes \xi + \frac{1}{|\xi |^2} \xi ^\perp \otimes \xi ^\perp$, one sees that the pair $(\Psi , \alpha )$ satisfies (and in fact is characterized uniquely by) the relation

$$\begin{aligned} \mathrm {D}\Phi (\xi ) + 2 \Psi (\xi ) \otimes \xi = \alpha (\xi ) \mathrm {Id} \, . \end{aligned}$$

(3.4)

Definition 3.2

Given $\Phi \in \mathrm {Ent}$, we define

$$\begin{aligned} \Vert \Phi \Vert _{\mathrm {Ent}} := \mathrm {Lip}(\Psi ), \end{aligned}$$

where $\mathrm {Lip}(\Psi )$ is the Lipschitz constant of the function $\Psi$ given by (3.3).

We remark that $\Vert \cdot \Vert _{\mathrm {Ent}}$ is a norm on $\mathrm {Ent}$. Indeed, $\Psi$ and $\alpha$ are linear in $\Phi$, see (3.3) and (3.2). Moreover, recalling that $\Phi$, $\Psi$, and $\alpha$ have compact support, if $\mathrm {Lip}(\Psi ) = 0$, then $\Psi \equiv 0$ and (3.4) yields $\mathrm {D}\Phi = \alpha \, \mathrm {Id}$. Since the row-wise $\mathrm {curl}(\alpha \, \mathrm {Id})$ equals $\nabla ^\perp \alpha$, we get $\alpha \equiv 0$ and thus $\Phi \equiv 0$.

Let $\mathcal {A}$ be the class of open and bounded subsets of $\mathbb {R}^2$ as in (2.15). To state our main result, we introduce the functional $H :L^1_{\mathrm {loc}}(\mathbb {R}^2;\mathbb {R}^2) {\times }\mathcal {A}\rightarrow [0,+\infty ]$ defined by

$$\begin{aligned} H(\chi ,\Omega ) := \bigvee _{\begin{array}{c} \Phi \in \mathrm {Ent}\\ \Vert \Phi \Vert _{\mathrm {Ent}} \leqq 1 \end{array}} | \mathrm {div}(\Phi \circ \chi ^\perp ) |(\Omega ) \, , \end{aligned}$$

(3.5)

if $\chi$ satisfies

$$\begin{aligned}&|\chi | = 1 \text { a.e. in } \Omega , \quad \mathrm {curl}(\chi ) = 0 \text { in } \mathcal {D}'(\Omega ), \quad \mathrm {div}(\Phi \circ \chi ^\perp ) \in \mathcal {M}_b(\Omega ) \nonumber \\&\quad \text { for all } \Phi \in \mathrm {Ent}, \end{aligned}$$

(3.6)

and H extended to $+\infty$ otherwise in $L^1_{\mathrm {loc}}(\mathbb {R}^2;\mathbb {R}^2)$. For a discussion on the role played by the functional H in the analysis of the classical Aviles–Giga functionals, we refer to Remark 3.5 below.

Using compactly supported instead of non-compactly supported entropies in the definition of H is not restrictive, as we show in Proposition 3.3 below. In particular, taking the supremum in (3.5) over the entropies introduced in [26] does not affect the values of the functional H.

Proposition 3.3

Let $\Phi \in C^\infty (\mathbb {R}^2 \setminus \{0\};\mathbb {R}^2)$ be a function satisfying (3.1) for $\xi \ne 0$. Notice that for such $\Phi$, (3.2), (3.3) define functions $\alpha \in C^\infty (\mathbb {R}^2 \setminus \{0\})$ and $\Psi \in C^\infty (\mathbb {R}^2 \setminus \{0\};\mathbb {R}^2)$. Assume that $\mathrm {Lip}(\Psi ) \leqq 1$. Let $\Omega \subset \mathbb {R}^2$ be an open and bounded set and let $\chi \in L^\infty (\Omega ;\mathbb {S}^1)$ satisfy $\mathrm {curl}(\chi ) = 0$ in $\mathcal {D}'(\Omega )$. Let $\Omega ' \subset \Omega$ be an open set. Then we have that

$$\begin{aligned} |\mathrm {div}(\Phi \circ \chi ^\perp )|(\Omega ') \leqq H(\chi ,\Omega '). \end{aligned}$$

Proof

We start by showing that the singularities of $\Phi , \Psi , \alpha$ at 0 can be removed. To this end we note that for $\Phi , \Psi , \alpha$ (3.4) holds true in $\mathbb {R}^2 \setminus \{0\}$. Computing the row-wise curl of both sides of this identity, and using that the curl of the identity $\xi \mapsto \xi$ vanishes, we get that

$$\begin{aligned} \nabla ^\perp \alpha (\xi ) = -2 \mathrm {D}\Psi (\xi ) \cdot \xi ^\perp . \end{aligned}$$

Since $\mathrm {Lip}(\Psi ) \leqq 1$, we obtain that $|\nabla \alpha (\xi )| \leqq 2 |\xi |$. Note that this implies that $\alpha$ is Lipschitz in $B_1(0) \setminus \{0\}$ and thus admits a unique continuous extension to the whole $\mathbb {R}^2$. In the same way, $\Psi$ admits a unique continuous extension to $\mathbb {R}^2$ which still satisfies $\mathrm {Lip}(\Psi ) \leqq 1$. By (3.4) we then infer that also $\mathrm {D}\Phi$ extends continuously to $\mathbb {R}^2$, and, as a consequence $\Phi$ can be extended to a $C^1$ function on the whole $\mathbb {R}^2$.

Next, we reduce the claim to the “effective entropy” $\Phi ^{\mathrm {eff}}$ defined on $\mathbb {R}^2$ by

$$\begin{aligned} \Phi ^{\mathrm {eff}}(\xi ) := \Phi (\xi ) - \Phi (0) - \alpha (0) \xi + |\xi |^2 \Psi (0). \end{aligned}$$

Observe that $\Phi ^{\mathrm {eff}}$ is $C^1$ on $\mathbb {R}^2$, smooth on $\mathbb {R}^2 \setminus \{0\}$ and satisfies (3.1). Since $|\chi | = 1$ and $\mathrm {curl}(\chi ) = 0$, we have that

$$\begin{aligned}&\mathrm {div}(\Phi ^{\mathrm {eff}} \circ \chi ^\perp ) = \mathrm {div}(\Phi \circ \chi ^\perp ) - \mathrm {div}(\Phi (0)) - \alpha (0) \mathrm {div}(\chi ^\perp ) + \mathrm {div}(|\chi |^2 \Psi (0) ) \nonumber \\&\quad = \mathrm {div}(\Phi \circ \chi ^\perp ). \end{aligned}$$

(3.7)

Moreover, the functions $\alpha ^{\mathrm {eff}}$ and $\Psi ^{\mathrm {eff}}$ associated to $\Phi ^{\mathrm {eff}}$ are given by

$$\begin{aligned} \alpha ^{\mathrm {eff}}(\xi ) = \alpha (\xi ) - \alpha (0) \quad \text {and} \quad \Psi ^{\mathrm {eff}}(\xi ) = \Psi (\xi ) - \Psi (0) \end{aligned}$$

and by our previous bound on $\nabla \alpha$ we infer that $|\alpha ^{\mathrm {eff}}(\xi )| \leqq C |\xi |^2$. We furthermore obtain the bounds

$$\begin{aligned} |\Psi ^{\mathrm {eff}}(\xi )| \leqq |\xi |, \quad |\mathrm {D}\Phi ^{\mathrm {eff}}(\xi )| \leqq C |\xi |^2, \quad |\Phi ^{\mathrm {eff}}(\xi )| \leqq C |\xi |^3 \end{aligned}$$

(3.8)

by recalling that $\mathrm {Lip}(\Psi ) \leqq 1$ and then using that (3.4) holds for $\Phi ^{\mathrm {eff}}, \Psi ^{\mathrm {eff}}, \alpha ^{\mathrm {eff}}$ and that $\Phi ^{\mathrm {eff}}(0) = 0$. Note moreover that $\mathrm {Lip}(\Psi ^{\mathrm {eff}}) \leqq 1$.

Let us now approximate $\Phi ^{\mathrm {eff}}$ by entropies $\Phi _k \in \mathrm {Ent}$. To this end we consider a sequence of functions $\zeta _k \in C_c^\infty ((0,\infty ))$ with $\zeta _k(1)=1$ and such that

$$\begin{aligned} 0 \leqq \zeta _k(s) \leqq 1, \quad |\zeta _k'(s)| \leqq \frac{C}{ks} \, , \quad |\zeta _k''(s)| \leqq \frac{C}{ks^2} \end{aligned}$$

(3.9)

for all $s > 0$, where the constant C is independent of k and s. To find the functions $\zeta _k$, we first construct a sequence of functions $\rho _k \in W^{2,\infty }((0, \infty ))$ with compact supports, satisfying the bounds in (3.9) and such that $\rho _k = 1$ in a neighborhood of 1. This can be achieved following the scheme shown in Fig. 3. The desired functions $\zeta _k$ are then obtained by mollifying $\rho _k$ on a sufficiently small scale.

Fig. 3

The figure shows the construction of $\rho _k'$. It is $\rho _k' = 0$ in $I_0 \cup I_2 \cup I_4$, $I_2$ being a neighborhood of 1. In $I_1$ and $I_3$, $\rho '_k$ takes the form of a hyperbolic arc and is given by $\pm \frac{1}{ks}$. In the four intervals in between, the pieces are joined together with hyperbolic arcs of the form $\pm \frac{2}{ks} + c$, where the constant c is chosen suitably for each individual interval. Notice that by positioning $I_1$ close enough to $s = 0$ and by letting $I_3$ extend far enough to the right, it is possible to achieve that both gray areas each have an area of 1. This is due to the fact that the integral of $\frac{1}{ks}$ is infinite both close to 0 and close to $\infty$. The primitive $\rho _k$ of $\rho _k'$ with $\rho _k(0) = 0$ has the desired properties

Let us define the approximations $\Phi _k \in C_c^\infty (\mathbb {R}^2 \setminus \{0\};\mathbb {R}^2)$ by

$$\begin{aligned} \Phi _k(\xi ) := \zeta _k(|\xi |) \Phi ^{\mathrm {eff}}(\xi ) \end{aligned}$$

and observe that they indeed satisfy (3.1). Let us estimate $\Vert \Phi _k \Vert _{\mathrm {Ent}}$. The function $\Psi _k$ associated to $\Phi _k$ through (3.2), (3.3) is given by

$$\begin{aligned} \Psi _k(\xi ) = \zeta _k(|\xi |) \Psi ^{\mathrm {eff}} (\xi ) - \frac{1}{2} \frac{\zeta _k'(|\xi |)}{|\xi |} \Phi ^{\mathrm {eff}}(\xi ). \end{aligned}$$

Moreover,

$$\begin{aligned} \mathrm {D}\Psi _k(\xi ) = \zeta _k(|\xi |) \mathrm {D}\Psi ^{\mathrm {eff}}(\xi ) + R_k(\xi ), \end{aligned}$$

where

$$\begin{aligned} R_k(\xi )&= \zeta _k'(|\xi |) \Psi ^{\mathrm {eff}}(\xi ) \otimes \frac{\xi }{|\xi |} - \frac{1}{2} \frac{\zeta _k'(|\xi |)}{|\xi |} \mathrm {D}\Phi ^{\mathrm {eff}}(\xi ) \\&\quad + \frac{1}{2} \frac{\zeta _k'(|\xi |)}{|\xi |^2} \Phi ^{\mathrm {eff}}(\xi ) \otimes \frac{\xi }{|\xi |} - \frac{1}{2} \frac{\zeta _k''(|\xi |)}{|\xi |} \Phi ^{\mathrm {eff}}(\xi ) \otimes \frac{\xi }{|\xi |}. \end{aligned}$$

By virtue of (3.8) and (3.9) we have that $R_k \rightarrow 0$ uniformly as $k \rightarrow \infty$. Since $|\zeta _k(|\xi |)| \leqq 1$ this implies that $\Vert \Phi _k \Vert _{\mathrm {Ent}} = \mathrm {Lip}(\Psi _k) \leqq \mathrm {Lip}(\Psi ^{\mathrm {eff}}) + o_k(1) \leqq 1 + o_k(1)$.

Finally, we show that

$$\begin{aligned} |\mathrm {div}(\Phi ^{\mathrm {eff}} \circ \chi ^\perp )|(\Omega ') \leqq (1+t) H(\chi ,\Omega ') \end{aligned}$$

for all $t > 0$. Indeed, choose k such that $\Vert \Phi _k \Vert _{\mathrm {Ent}} \leqq 1 + t$. The case $\Phi _k = 0$ being trivial, we may assume that $\Vert \Phi _k \Vert _{\mathrm {Ent}} > 0$. Then, since $\Phi ^{\mathrm {eff}} = \Phi _k$ on $\mathbb {S}^1$ we get that

$$\begin{aligned} |\mathrm {div}(\Phi ^{\mathrm {eff}} \circ \chi ^\perp )|(\Omega ') \leqq \Vert \Phi _k \Vert _{\mathrm {Ent}} \Big | \mathrm {div}\Big ( \tfrac{\Phi _k}{\Vert \Phi _k \Vert _{\mathrm {Ent}}} \circ \chi ^\perp \Big ) \Big |(\Omega ') \leqq (1+t) H(\chi ,\Omega '). \end{aligned}$$

In view of (3.7) this concludes the proof. $\square$

Remark 3.4

(Notions of entropy and the domain of the $\Gamma$-limit) Entropies are a central tool in the analysis of the Aviles–Giga functionals $AG_{\varepsilon }$ in (1.7). In this remark we give an overview of some notions of entropy in the context of Avilesp–Giga functionals available in the literature.

As explained above, our definition of entropies is inspired by that given in [26], where entropies are used to prove compactness properties of sequences with equibounded Aviles–Giga energies.

With the aim of better understanding the fine properties of solutions of the eikonal equation selected by the Aviles–Giga functionals, another definition of entropy has been given in [25]. There the authors explain that the asymptotic admissible set of the Aviles–Giga functionals is contained in the space $A(\Omega )$ of solutions to the eikonal equation $|\nabla \varphi | = 1$ satisfying

$$\begin{aligned} \mathrm {div}(\Phi \circ \nabla ^\perp \varphi ) \in \mathcal {M}_b(\Omega ) \end{aligned}$$

for all smooth $\Phi :\mathbb {S}^1 \rightarrow \mathbb {R}^2$ (the entropies in [25]) with the property that

$$\begin{aligned}&\text {if }U \subset \mathbb {R}^2 \text {is open, } m :U \rightarrow \mathbb {S}^1 \text { is smooth, and } \mathrm {div}(m) = 0, \nonumber \\&\quad \text {then } \mathrm {div}(\Phi \circ m) = 0. \end{aligned}$$

(3.10)

This notion of entropy (also used in other variants in [24, 28, 32, 35, 42]) and the one in Definition 3.1 (or in [26]) are basically equivalent. Specifically, every entropy $\Phi$ of the type (3.10) admits an extension to a smooth function on $\mathbb {R}^2$ that is an entropy in the sense of Definition 3.1. Conversely, for every entropy in the sense of Definition 3.1, its restriction to $\mathbb {S}^1$ satisfies (3.10). In particular, condition (3.6) for $\chi = \nabla \varphi$ is equivalent to requiring that $\varphi \in A(\Omega )$.

A smaller class of entropies has been considered in [5, 8, 34]. They are of the form

$$\begin{aligned} \Sigma _{\nu , \nu ^\perp }(\xi ) := \frac{2}{3} \big ( (\xi \cdot \nu ^\perp )^3 \nu + (\xi \cdot \nu )^3 \nu ^\perp \big ), \quad \nu \in \mathbb {S}^1. \end{aligned}$$

(3.11)

In [5] they are used to prove compactness of sequences with equibounded Aviles–Giga energy and to formulate an asymptotic lower bound (cf. Remark 3.5 below). In particular, it is shown that the asymptotic admissible set of the Aviles–Giga functionals is contained in the space $AG(\Omega )$ of solutions to the eikonal equation $|\nabla \varphi | = 1$ satisfying

$$\begin{aligned} \mathrm {div}(\Sigma _{\nu ,\nu ^\perp } \circ \nabla ^\perp \varphi ) \in \mathcal {M}_b(\Omega ) \end{aligned}$$

for all $\nu \in \mathbb {S}^1$ (in fact, it is equivalent to require this only for $\nu _1 = \big ( {\begin{matrix} 1 \\ 0 \end{matrix}} \big )$ and $\nu _2 = \frac{1}{\sqrt{2}} \big ( {\begin{matrix} 1 \\ 1 \end{matrix}} \big )$). As $\Sigma _{\nu ,\nu ^\perp }$ satisfy (3.10), the inclusion $A(\Omega ) \subset AG(\Omega )$ holds true.

To the best of our knowledge, it is not known whether $A(\Omega ) = AG(\Omega )$, i.e., whether all entropy productions $\mathrm {div}(\Phi \circ \nabla ^\perp \varphi )$ can be controlled by only the entropy productions $\mathrm {div}(\Sigma _{\nu ,\nu ^\perp } \circ \nabla ^\perp \varphi )$ if $\varphi$ solves $|\nabla \varphi | = 1$. This problem has been intensively studied in the recent years and several partial results have been obtained. As a first evidence, in [35, 37] it has been proved that if $\mathrm {div}(\Sigma _{\nu ,\nu ^\perp } \circ \nabla ^\perp \varphi ) = 0$ for $\nu = \nu _1, \nu _2$, then all entropy productions $\mathrm {div}(\Phi \circ \nabla ^\perp \varphi )$ vanish. In [37] this follows from the result that, under the previous assumption, $\nabla \varphi$ satisfies rigidity, i.e., $\nabla \varphi$ is locally Lipschitz outside a locally finite set of vortex-like singularities. In [24, 32] it is shown that also suitable fractional Sobolev regularity of $\nabla \varphi$ triggers the same rigidity. A further step towards understanding the threshold regularity for rigidity has been achieved in [28]. There it is shown that requiring that all entropy productions $\mathrm {div}(\Phi \circ \nabla \varphi )$ are finite measures is locally equivalent to the Besov regularity $\nabla \varphi \in B_{3,\infty }^{1/3}$. Already the stronger regularity $\nabla \varphi \in B_{3,q}^{1/3}$ for $q < \infty$ yields rigidity. In [38], the authors raise the question whether $B_{3p,\infty }^{1/3}$ regularity, $p > 1$, triggers this rigidity, too. As a partial result, they prove that this regularity implies that the entropy productions $\mathrm {div}(\Phi \circ \nabla ^\perp \varphi )$ belong to $L^p$, which they conjecture to be enough to deduce rigidity. Furthermore, the authors obtain further evidence that $\mathrm {div}(\Phi \circ \nabla ^\perp \varphi )$ can be controlled by $\mathrm {div}(\Sigma _{\nu ,\nu ^\perp } \circ \nabla ^\perp \varphi )$ for $\nu = \nu _1,\nu _2$. More precisely, it is shown that if $p \geqq \frac{4}{3}$, then $\mathrm {div}(\Sigma _{\nu ,\nu ^\perp } \circ \nabla ^\perp \varphi ) \in L^p$ implies that $\mathrm {div}(\Phi \circ \nabla ^\perp \varphi ) \in L^p$ for all entropies $\Phi$. Moreover, according to [41], a preliminary result on the question whether this can be extended to the case of measures is available as a consequence of recent developments, specifically, the eikonal equation’s kinetic formulation established in [28], a Lagrangian representation method [13, 39, 40, 42, 43], and ideas used in [38]. The precise statement requires the introduction of a subclass of parametrized entropies $\{ \Phi _f \ : \ f :\mathbb {S}^1 \rightarrow \mathbb {R}\}$ (cf. [28, Subsection 3.1]), which is rich enough to establish the kinetic formulation. The results in [28] imply that if $\mathrm {div}(\Phi _f \circ \nabla ^\perp \varphi ) \in \mathcal {M}_b$ for all f, then $\mathrm {div}(\Phi \circ \nabla ^\perp \varphi ) \in \mathcal {M}_b$ for all entropies $\Phi$. By [41], if it is assumed a priori that the entropy productions $\mathrm {div}(\Phi _f \circ \nabla ^\perp \varphi )$ are finite measures for all f, then the precise structure of the kinetic defect measure obtained in [42, Proposition 1.7] allows one to control $\mathrm {div}(\Phi _f \circ \nabla ^\perp \varphi )$, for all f, in terms of $\mathrm {div}(\Sigma _{\nu ,\nu ^\perp } \circ \nabla ^\perp \varphi )$, $\nu = \nu _1,\nu _2$, up to a multiplicative constant depending only on $\Phi _f$.

Remark 3.5

We introduce the functional H in (3.5) for the $\Gamma$-convergence analysis of the functionals $H_n$. In fact, $\varphi \mapsto H(\nabla \varphi , \Omega )$ is also a candidate for the $\Gamma$-limit of the classical Aviles–Giga functionals $AG_{\varepsilon }(\, \cdot , \Omega )$ defined (1.7). In particular, it can be shown that the liminf inequality holds true, i.e., that $\varphi _\varepsilon \rightarrow \varphi$ in $W^{1,1}_{\mathrm {loc}}(\Omega )$ implies that

$$\begin{aligned} \liminf _{\varepsilon \rightarrow 0} AG_\varepsilon (\varphi _\varepsilon , \Omega ) \geqq H(\nabla \varphi , \Omega ). \end{aligned}$$

(3.12)

We remark that all arguments required for the proof of this liminf inequality are contained in Section 6; we refer to Remark 6.1 for an outline of the proof.³

We remark that in the analysis of the Aviles–Giga functionals $AG_\varepsilon$ the candidate $\Gamma$-limit most often used in the literature is given by $\varphi \mapsto H^0(\nabla \varphi , \Omega )$, where $H^0$ slightly differs from (3.5). More specifically,

$$\begin{aligned} H^0(\chi ,\Omega ) := \bigvee _{\nu \in \mathbb {S}^1} | \mathrm {div}(\Sigma _{\nu , \nu ^\perp } \circ \chi ^\perp ) | (\Omega ) = \left| \begin{pmatrix} \mathrm {div}(\Sigma _{\nu _1, \nu _1^\perp } \circ \chi ^\perp ) \\ \mathrm {div}(\Sigma _{\nu _2, \nu _2^\perp } \circ \chi ^\perp ) \end{pmatrix} \right| (\Omega ).\nonumber \\ \end{aligned}$$

(3.13)

Here, $\nu _1 = \big ( {\begin{matrix} 1 \\ 0 \end{matrix}} \big )$, $\nu _2 = \frac{1}{\sqrt{2}} \big ( {\begin{matrix} 1 \\ 1 \end{matrix}} \big )$ and $\Sigma _{\nu , \nu ^\perp }$ are the entropies defined by (3.11). The functional $H^0(\chi ,\Omega )$ is defined by (3.13) if $\chi$ satisfies

$$\begin{aligned}&|\chi | = 1 \text { a.e. in } \Omega , \quad \mathrm {curl}(\chi ) = 0 \text { in } \mathcal {D}'(\Omega ), \quad \mathrm {div}(\Sigma _{\nu ,\nu ^\perp } \circ \chi ^\perp ) \in \mathcal {M}_b(\Omega )\nonumber \\&\quad \text { for all } \nu \in \mathbb {S}^1, \end{aligned}$$

(3.14)

and extended to $+ \infty$ otherwise. The functional $H^0$ has first been considered in [5, 8], where it has been shown that $\varphi \mapsto H^0(\nabla \varphi , \Omega )$ provides a lower bound on the $\Gamma \text {-} \liminf$ of the Aviles–Giga functionals $AG_\varepsilon (\, \cdot , \Omega )$. As it is still not known whether the domain of $H^0$ is contained in $A(\Omega )$ (cf. Remark 3.4), it is natural to look for a limit functional that takes into account all entropy productions, such as H in (3.5).

Let us discuss next why the lower bound (3.12) is coherent with the already known results on the $\Gamma$-limiting behavior of the Aviles–Giga functionals. In Corollary 3.6 below, we show that $H \geqq H^0$. For a discussion about whether $H = H^0$, see Remark 3.4 above. Since $H \geqq H^0$, H provides a lower bound of the $\Gamma \text {-} \liminf AG_\varepsilon$ that is possibly sharper than $H^0$. Moreover, in Corollary 3.8 below we show that

$$\begin{aligned}&\chi \in BV(\Omega ;\mathbb {S}^1) \text { and } \mathrm {curl}(\chi ) = 0 \quad \implies \quad H(\chi ,\Omega ) = H^0(\chi , \Omega ) \nonumber \\&\quad = \frac{1}{6} \int _{J_\chi } |[\chi ]|^3 \, {\mathrm {d}{\mathcal {H}}}^1. \end{aligned}$$

(3.15)

In particular, the lower bound obtained from H is optimal on $\varphi$ if $\nabla \varphi \in BV(\Omega ;\mathbb {S}^1)$, as for such $\varphi$ the limsup inequality corresponding to $H^0$ has been proved in [22, 46].

As we show in Proposition 3.7 below, the theory established in [25] allows us to prove that, even if $\chi$ is not BV, the restriction of $H(\chi , \, \cdot \, )$ to the jump set $J_\chi$ is still given by $\frac{1}{6} \int _{J_\chi } |[\chi ]|^3 \, {\mathrm {d}{\mathcal {H}}}^1$. It is however not known whether H is concentrated on $J_\chi$. This is related to a conjecture raised in [25, Conjecture 1], which would imply that the identity $H(\chi ,\Omega ) = \frac{1}{6} \int _{J_\chi } |[\chi ]|^3 \, {\mathrm {d}{\mathcal {H}}}^1$ holds for all $\chi$ satisfying (3.6). We remark that concentration results of this kind have been proved for related models in [40, 43].

The following result is a consequence of Proposition 3.3.

Corollary 3.6

Let H be the functional in (3.5) and let $H^0$ be defined by (3.13). We have that $H \geqq H^0$.

Proof

For $\nu \in \mathbb {S}^1$, we compute the derivative of the function $\Sigma _{\nu ,\nu ^\perp }$ defined by (3.11) to be

$$\begin{aligned} \mathrm {D}\Sigma _{\nu ,\nu ^\perp }(\xi ) = 2 \big ( (\xi \cdot \nu ^\perp )^2 \nu \otimes \nu ^\perp + (\xi \cdot \nu )^2 \nu ^\perp \otimes \nu \big ) \, . \end{aligned}$$

Using the elementary identites $\xi ^\perp \cdot \nu ^\perp = \xi \cdot \nu$ and $\xi ^\perp \cdot \nu = - \xi \cdot \nu ^\perp$ we obtain that $\Sigma _{\nu ,\nu ^\perp }$ satisfies (3.1). Computing the functions $\alpha$ and $\Psi$ associated to $\Sigma _{\nu ,\nu ^\perp }$ through (3.2), (3.3), we obtain

$$\begin{aligned} \alpha (\xi )&= -2 (\xi \cdot \nu ^\perp )(\xi \cdot \nu ), \\ \Psi (\xi )&= - (\xi \cdot \nu ^\perp )\nu - (\xi \cdot \nu )\nu ^\perp = - (\nu ^\perp \otimes \nu + \nu \otimes \nu ^\perp ) \xi , \end{aligned}$$

where we have used the identities $|\xi |^2 = (\xi \cdot \nu )^2 + (\xi \cdot \nu ^\perp )^2$ and $\xi = (\xi \cdot \nu )\nu + (\xi \cdot \nu ^\perp )\nu ^\perp$. Since the matrix $\nu ^\perp \otimes \nu + \nu \otimes \nu ^\perp$ is orthogonal, we find that $\mathrm {Lip}(\Psi ) = 1$. As a consequence, applying Proposition 3.3 to $\Sigma _{\nu ,\nu ^\perp }$, we get for every $\chi$ satisfying (3.14) that

$$\begin{aligned} |\mathrm {div}(\Sigma _{\nu ,\nu ^\perp } \circ \chi ^\perp )|(\Omega ') \leqq H(\chi ,\Omega ') \end{aligned}$$

for every open $\Omega ' \subset \Omega$. By considering partitions of $\Omega$ to pass to the supremum, we then infer that $H^0(\chi , \Omega ') \leqq H(\chi , \Omega )$ as desired. $\square$

For the next result we recall that the jump set $J_v$ is defined for every $v \in L^1_{\mathrm {loc}}(\Omega ;\mathbb {R}^2)$ according to Section 2.3.

Proposition 3.7

Let $\Omega \subset \mathbb {R}^2$ be an open and bounded set and let $\chi \in L^1_{\mathrm {loc}}(\mathbb {R}^2;\mathbb {R}^2)$ satisfy (3.6). Let $J_\chi$ be the jump set of $\chi |_\Omega$. Then we have that

$$\begin{aligned} \bigvee _{\begin{array}{c} \Phi \in \mathrm {Ent}\\ \Vert \Phi \Vert _{\mathrm {Ent}} \leqq 1 \end{array}} | \mathrm {div}(\Phi \circ \chi ^\perp ) |(J_\chi ) = \frac{1}{6} \int _{J_\chi } |[\chi ]|^3 \, {\mathrm {d}{\mathcal {H}}}^1. \end{aligned}$$

Proof

Due to the relation between entropies in $\mathrm {Ent}$ and functions $\Phi$ satisfying (3.10) as explained in Remark 3.4, the theory in [25] and specifically [25, Theorem 1] applies to $\chi$. (More precisely, as the authors in [25] work with divergence-free fields instead of curl-free fields, we apply their results to $\chi ^\perp$.) According to this theory, there exists a set $J \subset \Omega$, coinciding with $J_\chi$ up to a $\mathcal {H}^1$-null set, such that

for all $\Phi :\mathbb {S}^1 \mapsto \mathbb {R}^2$ satisfying (3.10). As a consequence,

Since the restriction to $\mathbb {S}^1$ of any $\Phi \in \mathrm {Ent}$ satisfies (3.10), the above equation is also true for every $\Phi \in \mathrm {Ent}$. The same applies to $\Phi = \Sigma _{\nu ,\nu ^\perp }$ for any $\nu \in \mathbb {S}^1$ as well. As a consequence we have that

$$\begin{aligned} \mu (J_\chi ) = \int _{J_\chi } \sup _{\begin{array}{c} \Phi \in \mathrm {Ent}\\ \Vert \Phi \Vert _{\mathrm {Ent}} \leqq 1 \end{array}} \Big | \Big ( \Phi \left( (\chi ^\perp )^+ \right) - \Phi \left( (\chi ^\perp )^- \right) \Big ) \cdot \nu _\chi \Big | \, {\mathrm {d}{\mathcal {H}}}^1 \end{aligned}$$

(3.16)

and

$$\begin{aligned} \mu ^0(J_\chi ) = \int _{J_\chi } \sup _{\nu \in \mathbb {S}^1} \Big | \Big ( \Sigma _{\nu ,\nu ^\perp } \left( (\chi ^\perp )^+ \right) - \Sigma _{\nu ,\nu ^\perp } \left( (\chi ^\perp )^- \right) \Big ) \cdot \nu _\chi \Big | \, {\mathrm {d}{\mathcal {H}}}^1, \end{aligned}$$

(3.17)

where we have set

$$\begin{aligned} \mu := \bigvee _{\begin{array}{c} \Phi \in \mathrm {Ent}\\ \Vert \Phi \Vert _{\mathrm {Ent}} \leqq 1 \end{array}} | \mathrm {div}(\Phi \circ \chi ^\perp ) | \quad \text {and} \quad \mu ^0 := \bigvee _{\nu \in \mathbb {S}^1} | \mathrm {div}(\Sigma _{\nu ,\nu ^\perp } \circ \chi ^\perp ) |. \end{aligned}$$

Let us note that from Corollary 3.6 it follows that $\mu \geqq \mu ^0$. Let us also note that from $|\chi | = 1$ a.e. it follows that $\chi ^+(x), \chi ^-(x) \in \mathbb {S}^1$ for every $x \in J_\chi$. Let us fix $x \in J_\chi$. We recall from Section 2.3 that there exists a $d{} \in \mathbb {R}$ such that $\chi ^+(x) - \chi ^-(x) = d{} \, \nu _\chi (x)$.

We now claim that for all $a,b \in \mathbb {S}^1$ and $\nu \in \mathbb {S}^1$ with the properties that $a \ne b$ and $(a - b) = d{} \, \nu$ for some $d{} \in \mathbb {R}$, we have that

$$\begin{aligned} \big | \big ( \Phi ( a^\perp ) - \Phi ( b^\perp ) \big ) \cdot \nu \big | \leqq \frac{1}{6} |a-b|^3 \quad \text {for all } \Phi \in \mathrm {Ent}\text { with } \Vert \Phi \Vert _{\mathrm {Ent}} \leqq 1\nonumber \\ \end{aligned}$$

(3.18)

and

$$\begin{aligned} \big | \big ( \Sigma _{\nu ,\nu ^\perp } ( a^\perp ) - \Sigma _{\nu ,\nu ^\perp } ( b^\perp ) \big ) \cdot \nu \big | = \frac{1}{6} |a-b|^3. \end{aligned}$$

(3.19)

As a consequence, the supremum in (3.17) at x is attained for $\nu = \nu _\chi (x)$, takes the value $\frac{1}{6} |[\chi ](x)|^3$, and coincides with the supremum in (3.16) at x. This concludes the proof.

To prove (3.18), let us note that the conditions on $a,b,\nu ,d{}$ imply that $a \cdot \nu ^\perp = b \cdot \nu ^\perp \in \{ \pm \sqrt{1 - |d{}|^2/4} \}$ and $a \cdot \nu = - b \cdot \nu = \frac{d{}}{2}$. For $\Phi \in \mathrm {Ent}$ with $\Vert \Phi \Vert _{\mathrm {Ent}} \leqq 1$ we get that

$$\begin{aligned} \begin{aligned}&\big ( \Phi ( a^\perp ) - \Phi ( b^\perp ) \big ) \cdot \nu \\&\quad = \int _{0}^{\frac{d{}}{2}} \nu \cdot \frac{\mathrm {d}}{\mathrm {d}s} \Big ( \Phi \big (- (a \cdot \nu ^\perp ) \nu + s \nu ^\perp \big ) - \Phi \big (- (a \cdot \nu ^\perp ) \nu - s \nu ^\perp \big ) \Big )\, {\mathrm {d}{s}} \\&\quad = -2 \int _{0}^{\frac{d{}}{2}} \nu \cdot \Big ( \Psi \big (- (a \cdot \nu ^\perp ) \nu + s \nu ^\perp \big ) - \Psi \big (- (a \cdot \nu ^\perp ) \nu - s \nu ^\perp \big ) \Big ) \, s \, {\mathrm {d}{s}} \end{aligned} \end{aligned}$$

where we have used that (3.4) yields $\nu \cdot (\mathrm {D}\Phi (\xi ) \nu ^\perp ) = -2 (\nu \cdot \Psi (\xi )) (\xi \cdot \nu ^\perp )$, $\Psi$ being the function associated to $\Phi$ through (3.3). By Definition 3.2 we have that $\mathrm {Lip}(\Psi ) \leqq 1$ and thus we infer that

$$\begin{aligned} \big | \big ( \Phi ( a^\perp ) - \Phi ( b^\perp ) \big ) \cdot \nu \big | \leqq 2 \int _0^{\frac{|d{}|}{2}} 2s^2 \, {\mathrm {d}{s}} = \frac{4}{3} \frac{|d{}|^3}{8} = \frac{1}{6} |a-b|^3 \end{aligned}$$

as desired.

To prove (3.19), from the definition of $\Sigma _{\nu ,\nu ^\perp }$ in (3.11) we compute that

$$\begin{aligned} \big | \big ( \Sigma _{\nu ,\nu ^\perp } ( a^\perp ) - \Sigma _{\nu ,\nu ^\perp } ( b^\perp ) \big ) \cdot \nu \big |= & {} \frac{2}{3} \big | (a^\perp \cdot \nu ^\perp )^3 - (b^\perp \cdot \nu ^\perp )^3 \big | \\= & {} \frac{2}{3} \frac{|d{}|^3}{4} = \frac{1}{6} |a-b|^3, \end{aligned}$$

where we have used that $a \cdot \nu = - b \cdot \nu = \frac{d{}}{2}$. $\square$

Corollary 3.8

Let $\Omega$, $\chi$, and $J_\chi$ be as in Proposition 3.7. If additionally $\chi \in BV(\Omega ;\mathbb {S}^1)$, then we have that

$$\begin{aligned} H(\chi , \Omega ) = \frac{1}{6} \int _{J_\chi } |[\chi ]|^3 \, {\mathrm {d}{\mathcal {H}}}^1. \end{aligned}$$

Proof

By Proposition 3.7 and the definition (3.5), it remains only to prove that $|\mathrm {div}(\Phi \circ \chi ^\perp )|(\Omega \setminus J_\chi ) = 0$ for every $\Phi \in \mathrm {Ent}$. Fix $\Phi \in \mathrm {Ent}$, let $\Psi$ be defined by (3.3), and let us set ${\widetilde{\Phi }}(\xi ) := \Phi (\xi ) - (1 - |\xi |^2) \Psi (\xi )$. We observe that $\Phi \circ \chi ^\perp = {\widetilde{\Phi }} \circ \chi ^\perp$ a.e. in $\Omega$. Moreover, ${\widetilde{\Phi }} \in C_c^\infty (\mathbb {R}^2 \setminus \{0\}; \mathbb {R}^2)$ and therefore, by the Vol’pert chain rule (cf. Section 2.2), we have that

$$\begin{aligned} |\mathrm {div}(\Phi \circ \chi ^\perp )|(\Omega \setminus J_\chi )= & {} |\mathrm {div}({\widetilde{\Phi }} \circ \chi ^\perp )|(\Omega \setminus J_\chi ) \\= & {} \big |\mathrm {tr} \big ( \mathrm {D}{\widetilde{\Phi }}(\chi ^\perp ) (\mathrm {D}^a \chi ^\perp + \mathrm {D}^c \chi ^\perp ) \big ) \big | (\Omega ). \end{aligned}$$

Recall that in the above formula, $\mathrm {D}{\widetilde{\Phi }}$ is evaluated at the approximate limits of $\chi ^\perp$. Since $\chi ^\perp \in \mathbb {S}^1$ a.e. in $\Omega$, its approximate limit lies in $\mathbb {S}^1$ at every point where it is defined. Next, observe that $\mathrm {D}{\widetilde{\Phi }}(\xi ) = \alpha (\xi ) \mathrm {Id} - (1 - |\xi |^2) \mathrm {D}\Psi (\xi )$ by (3.4). As a consequence,

$$\begin{aligned} \mathrm {tr} \big ( \mathrm {D}{\widetilde{\Phi }}(\chi ^\perp ) (\mathrm {D}^a \chi ^\perp + \mathrm {D}^c \chi ^\perp ) \big ) = \alpha (\chi ^\perp ) \mathrm {tr} (\mathrm {D}^a \chi ^\perp + \mathrm {D}^c \chi ^\perp ) = 0, \end{aligned}$$

since $\mathrm {curl}(\chi ) = 0$ implies that the absolutely continuous and Cantor parts of $\mathrm {div}(\chi ^\perp )$ vanish. This concludes the proof. $\square$

4 Statement of the Main Results

4.1 List of variables, parameters, and symbols

For the reader’s convenience we summarize in the following list the main variables and parameters used in the paper:

• $\lambda _n$ is the lattice spacing. We assume that $\lambda _n\rightarrow 0$.

• $\alpha _n$ is the parameter in the energy (1.1) depending on $\lambda _n$. We assume that $\alpha _n\rightarrow 0$. Moreover, $\beta _n\equiv 2$.

• $\delta _n:= 4 - \frac{\alpha _n}{2}$ is set to get the identities (2.10)–(2.11). We have that $\delta _n\rightarrow 0$;

• $\varepsilon _n:= \frac{\lambda _n}{\sqrt{\delta _n}}$ is the parameter corresponding to the parameter $\varepsilon$ in the analogy between the energies $H_n$ and the Aviles–Giga functionals $AG_\varepsilon$ in (1.7). We assume that $\varepsilon _n\rightarrow 0$.

• We let $u \in \mathcal {PC}_{\lambda _n}(\mathbb {S}^1)$ denote spin fields, interpreted as $\mathbb {S}^1$-valued piecewise constant functions.

• $\theta ^{\mathrm {hor}}$ and $\theta ^{\mathrm {ver}}$ are the oriented angles between adjacent spins of the spin field u as defined in (2.6).

• $\chi$ is the relevant variable for the main result in the paper. It is defined in terms of $\theta ^{\mathrm {hor}}$ and $\theta ^{\mathrm {ver}}$ in (2.8) and represents the direction along which the helical configuration is rotating most, see Fig. 2.

• ${\widetilde{\chi }}$ is a variant of $\chi$ defined in (2.8).

• $\overline{\chi }$ is the linearized variant of $\chi$ defined in (2.19). As $n \rightarrow \infty$ we heuristically have that $\chi \simeq {\widetilde{\chi }} \simeq \overline{\chi }$.

• $\mathcal {A}_0$ is the class of admissible domains $\Omega$ in our problem defined by (2.14).

• $H_n$ are the discrete functionals studied in this paper and defined by (2.16).

• $W^\mathrm {d}$ and $A^\mathrm {d}$ are discrete operators used to define $H_n$. They are defined in (2.12).

• $H_n^*$ are the auxiliary Aviles–Giga-like discrete functionals defined by (2.22), which help in providing bounds on $\chi$ through Proposition 2.6.

• W is the potential in the classical Aviles–Giga functionals, $W(\xi ) = (1- |\xi |^2)^2$.

• H is the candidate discrete-to-continuum $\Gamma$-limit of the energies $H_n$. It is defined in (3.5).

• $\mathrm {Ent}$ is the space of entropies defined in Definition 3.1 and $\Vert \cdot \Vert _{\mathrm {Ent}}$ is a norm on $\mathrm {Ent}$ defined by Definition 3.2.

4.2 The main result

We state here the main result in the paper.

Theorem 4.1

Let $\Omega \in \mathcal {A}_0$. The following results hold true:

• (i) (Compactness) Let $(\chi _n)_n \in L^1_{\mathrm {loc}}(\mathbb {R}^2;\mathbb {R}^2)$ be a sequence such that

  $$\begin{aligned} \sup _n H_n(\chi _n,\Omega ) < +\infty . \end{aligned}$$

  Then there exists $\chi \in L^\infty (\mathbb {R}^2;\mathbb {S}^1)$ solving

  $$\begin{aligned} |\chi | = 1 \text { a.e. in } \Omega , \quad \mathrm {curl}(\chi ) = 0 \text { in } \mathcal {D}'(\Omega ), \end{aligned}$$

  (4.1)

  such that, up to a subsequence, $\chi _n \rightarrow \chi$ in $L^p_{\mathrm {loc}}(\Omega ;\mathbb {R}^2)$ for every $p \in [1,6)$.

• (ii) (liminf inequality) Let $(\chi _n)_n, \chi \in L^1_{\mathrm {loc}}(\mathbb {R}^2;\mathbb {R}^2)$ be such that $\chi _n \rightarrow \chi$ in $L^1_{\mathrm {loc}}(\Omega ;\mathbb {R}^2)$. Then

  $$\begin{aligned} H(\chi ,\Omega ) \leqq \liminf _{n} H_n(\chi _n,\Omega ). \end{aligned}$$

  (4.2)

• (iii) (limsup inequality) Assume that $\frac{\delta _n^{5/2}}{\lambda _n} \rightarrow 0$ as $n \rightarrow \infty$. Let $\chi \in L^1_{\mathrm {loc}}(\mathbb {R}^2;\mathbb {R}^2)$. Assume additionally that $\chi \in BV(\Omega ;\mathbb {R}^2)$. Then there exists a sequence $(\chi _n)_n \in L^1_{\mathrm {loc}}(\mathbb {R}^2;\mathbb {R}^2)$ such that $\chi _n \rightarrow \chi$ in $L^1(\Omega ;\mathbb {R}^2)$ and

  $$\begin{aligned} \limsup _{n} H_n(\chi _n,\Omega ) \leqq H(\chi ,\Omega ). \end{aligned}$$

  More precisely, if $H(\chi ,\Omega ) < +\infty$, then $\chi \in L^\infty (\Omega ;\mathbb {S}^1)$ and the recovery sequence $(\chi _n)_n$ is bounded in $L^\infty (\mathbb {R}^2;\mathbb {R}^2)$ and satisfies $\chi _n \rightarrow \chi$ in $L^p(\Omega ;\mathbb {R}^2)$ for every $p \in [1,\infty )$.

Remark 4.2

Note that, if $\sup _n H_n(\chi _n) < +\infty$, then Theorem 4.1-i) implies that there is a subsequence (not relabeled) such that $\chi _n \rightarrow \chi$ in $L^p_{\mathrm {loc}}(\Omega ;\mathbb {R}^2)$ for every $p \in [1,6)$, and $\chi$ satisfies the eikonal equation (4.1). Additionally, by Theorem 4.1-ii) we deduce that $H(\chi ) < +\infty$, namely $\chi$ satisfies

$$\begin{aligned} \mathrm {div}(\Phi \circ \chi ^\perp ) \in \mathcal {M}_b(\Omega ) \text { for all } \Phi \in \mathrm {Ent}, \quad \end{aligned}$$

and $\bigvee \{ | \mathrm {div}(\Phi \circ \chi ^\perp ) | : \Phi \in \mathrm {Ent}, \ \Vert \Phi \Vert _{\mathrm {Ent}} \leqq 1\}$ is a finite measure. Hence, $\chi$ is a (strong) finite entropy production solution of the eikonal equation (cf. [28, Definition 2.3] for a similar definition).

Remark 4.3

The proof of the compactness Theorem 4.1-i) as well as that of the liminf inequality Theorem 4.1-ii) do not require the simple connectedness of $\Omega$ and the regularity of its boundary.

Remark 4.4

Our $\Gamma$-convergence result is partial in that the limsup inequality requires that $\chi$ is BV and the additional scaling assumption $\frac{\delta _n^{5/2}}{\lambda _n} \rightarrow 0$. The former assumption reflects the fact that the limsup inequality for the classical Aviles–Giga functionals is only known for BV fields, cf. [22, 46]. Improving Theorem 4.1-iii) by only requiring that $\chi$ is such that $H(\chi ,\Omega ) < +\infty$ is out of the scope of this paper and it requires new developments in the analysis of the Aviles–Giga functionals.

The scaling assumption $\frac{\delta _n^{5/2}}{\lambda _n} \rightarrow 0$ is technical. It is due to the fact that the variable $\chi _n$, which enters the potential $W^\mathrm {d}$ in our energy $H_n$, is not equal to the curl-free variable $\overline{\chi }_n$ that we use in our construction of the recovery sequence. In the energy we therefore commit a bulk error (that is, away from the jump set $J_\chi$, where all of the asymptotic energy H concentrates). The scaling assumption is needed to control this bulk error.

We remark that we do not require an additional scaling assumption in our liminf inequality, as we are able to solve the mentioned problem in this case, through the introduction of approximate entropies (cf. (6.5)–(6.8) and Lemma 6.3).

We finally remark that, in terms of $\lambda _n$ and $\varepsilon _n$ the scaling assumption $\frac{\delta _n^{5/2}}{\lambda _n} \rightarrow 0$ can be read as an additional assumption on the asymptotic relation $\lambda _n\ll \varepsilon _n$. Indeed, the scaling assumption is satisfied whenever $\frac{\lambda _n^4}{\varepsilon _n^5} \rightarrow 0$, e.g., if $\lambda _n= \varepsilon _n^p$ with $p > \frac{5}{4}$.

Remark 4.5

The $\Gamma$-convergence analysis carried out for the functionals $H_n$ to prove Theorem 4.1 can be applied with minor modifications also to the discrete Aviles–Giga functionals $AG^{\mathrm {d}}_n$ defined by (2.23) in the regime $\frac{\lambda _n}{\varepsilon _n} \rightarrow 0$ as $n \rightarrow \infty$. Hence, the analogous results as in Theorem 4.1 can be proved for the functionals $AG^{\mathrm {d}}_n$, too. Moreover, in many cases our arguments can be simplified as we explain in Remarks 5.3, 6.2, and 7.4. In particular, we stress that the analogue of the limsup inequality in Theorem 4.1-iii) holds true for the functionals $AG^{\mathrm {d}}_n$ without the additional scaling assumption $\frac{\delta _n^{5/2}}{\lambda _n} \rightarrow 0$ (where $\delta _n= \frac{\lambda _n}{\varepsilon _n^2}$).

Remark 4.6

We recall that the functionals $H_n$ represent the behavior of the $J_1$–$J_2$–$J_3$ energies $F_n$ close to the helimagnet/ferromagnet transition point ($\alpha _n- (4 + 2 \beta _n) \nearrow 0$) if the next-to-nearest neighbors interaction parameter $\beta _n$ is chosen as $\beta _n\equiv 2$. We collect here some remarks about the cases where $0 \leqq \beta _n< 2$. Setting $\delta _n:= 4 - \frac{2 \alpha _n}{2 + \beta _n}$ and rescaling (1.2), a computation similar to (2.9)–(2.11) shows that the rescaled energy $\frac{1}{\delta _n^{3/2} \lambda _n} F_n$ is given by the convex combination $\frac{\beta _n}{2} H_n^{(2)} + \big ( 1 - \frac{\beta _n}{2} \big ) H_n^{(0)}$. Here, $H_n^{(2)}$ is given by the same expression as $H_n$ in (2.16) (with $\varepsilon _n$ adapted using $\delta _n= 4 - \frac{2 \alpha _n}{2 + \beta _n}$) and shares the same compactness properties. Moreover, $H_n^{(0)}$ corresponds to the $J_1$–$J_3$ energy studied in [17]. The observations therein show that $H_n^{(0)}$ takes the form

$$\begin{aligned} H_n^{(0)}(\chi ) = \frac{1}{2} \int _{\Omega } \frac{1}{\varepsilon _n} W^\mathrm {d}_{(0)}(\chi ) + \varepsilon _n|A^\mathrm {d}_{(0)}(\chi )|^2 \, {\mathrm {d}{x}}, \end{aligned}$$

where $W^\mathrm {d}_{(0)}(\chi ) \simeq (\frac{1}{2} - |\chi _1|^2)^2 + (\frac{1}{2} - |\chi _2|^2)^2$ and $A^\mathrm {d}_{(0)}(\chi ) \simeq \big ( |\partial ^{\mathrm {d}}_1 \chi _1|^2 + |\partial ^{\mathrm {d}}_2 \chi _2|^2 \big )^{1/2}$ as $n \rightarrow \infty$. Although similar in form to $H_n^{(2)}$, the behavior of this energy is very different from that of $H_n^{(2)}$. Indeed, its compactness is substantially stronger as it allows the values of the limit $\chi$ only to lie in four isolated points and, moreover, $\chi \in BV$, cf. [17, Theorem 2.1-i)].

The analysis in [17] together with the analysis carried out in this paper, allows us to understand the compactness properties of the rescaled $F_n$ for general $\beta _n\in [0,2]$ as a combination of the compactness properties of $H_n^{(0)}$ and $H_n^{(2)}$.

In the case that $\sup _n \beta _n< 2$, a bound on the energies $\frac{1}{\delta _n^{3/2} \lambda _n} F_n$ implies a bound on $H_n^{(0)}$. Since moreover $H_n^{(2)}$ can be controlled by $H_n^{(0)}$ up to a multiplicative constant, in this case the compactness of the rescaled $F_n$ is the same as in the $J_1$–$J_3$ model, cf. also [17, Remark 2.3].

If instead $\beta _n\rightarrow 2$, a bound on $\frac{1}{\delta _n^{3/2} \lambda _n} F_n$ implies only a bound on $H_n^{(2)}$ and on $\big (1-\frac{\beta _n}{2} \big ) H_n^{(0)}$. The question whether the latter term improves the compactness of the energy $H_n^{(2)}$ as in Theorem 4.1-i), ii) depends on the relative speed of the convergences $\beta _n\rightarrow 2$ and $\varepsilon _n\rightarrow 0$. In the case that $\frac{2 - \beta _n}{\varepsilon _n} \leqq C$, no improved compactness can be expected. Indeed, it can be observed that $|A^\mathrm {d}_{(0)} (\chi )|^2 \leqq C |\mathrm {D}^{\mathrm {d}}\overline{\chi }|^2$ and that

$$\begin{aligned} \sup _n \int _{\Omega } W(\chi _n) \, \, {\mathrm {d}{x}}< + \infty \quad \implies \quad \sup _n \int _{\Omega } W^\mathrm {d}_{(0)}(\chi _n) \, \, {\mathrm {d}{x}} < + \infty \end{aligned}$$

for all $\chi _n = \chi (u_n)$, $u_n \in \mathcal {PC}_{\lambda _n}(\mathbb {S}^1)$. As a consequence, it can be seen that a uniform bound on $H_n^{(2)}(\chi _n, \Omega )$ already implies (locally in $\Omega$) a uniform bound on $\big (1-\frac{\beta _n}{2} \big ) H_n^{(0)}(\chi )$ through Proposition 2.6 and (2.24).

However, if $\frac{2 - \beta _n}{\varepsilon _n} \rightarrow + \infty$, then the bound $C \geqq \big (1-\frac{\beta _n}{2} \big ) H_n^{(0)}(\chi _n) \geqq \frac{2 - \beta _n}{2 \varepsilon _n} \int _{\Omega } W^\mathrm {d}_{(0)}(\chi _n) \, {\mathrm {d}{x}}$ implies that the limit $\chi$ (obtained from the compactness of $H_n^{(2)}$) satisfies $\chi (x) \in \big \{ \pm \frac{1}{\sqrt{2}} \big \}^2$ for a.e. x. In particular, it attains only finitely many values and by Proposition 4.7 below we obtain $\chi \in BV \big (\Omega ;\big \{ \pm \frac{1}{\sqrt{2}} \big \}^2 \big )$. Thus, a posteriori the stronger compactness of the $J_1$–$J_3$ model is recovered.

Proposition 4.7

Let $\chi \in L^\infty (\Omega ;\mathbb {S}^1)$ be such that $H(\chi , \Omega ) < + \infty$. If $\chi$ attains values in a finite set a.e., then $\chi \in BV(\Omega ;\mathbb {S}^1)$.

Proof

We recall that thanks to [28, Theorem 2.6], $\chi$ being a finite entropy production solution implies that $\chi \in B_{3, \infty }^{1/3}(\Omega ')$ for all open sets $\Omega ' \subset \subset \Omega$. (As in [28] the authors work with divergence-free fields, we apply their results to $\chi ^\perp$.) Accordingly, (cf. also [36, Definition 14.1])

$$\begin{aligned} \sup _{t > 0} \sup _{|z| \leqq t} \int _{\Omega ' \cap (\Omega ' - z)} \frac{|\chi (x + z) - \chi (x)|^3}{t} \, {\mathrm {d}{x}} < + \infty . \end{aligned}$$

Since $\chi$ takes only finitely many values, we find a constant C such that $|\chi (x + z) - \chi (x)| \leqq C |\chi (x + z) - \chi (x)|^3$ for a.e. $x \in \Omega$. As a consequence $\sup _{z \ne 0} \int _{\Omega ' \cap (\Omega ' - z)} \frac{|\chi (x + z) - \chi (x)|}{|z|} \, {\mathrm {d}{x}} < + \infty$, which implies that $\chi \in BV_{\mathrm {loc}}(\Omega ';\mathbb {S}^1)$ (cf. [36, Theorem 13.48]). Applying now Corollary 3.8 locally in $\Omega$, we obtain that

$$\begin{aligned} \sup _{\Omega ' \subset \subset \Omega } \int _{J_\chi \cap \Omega '} |[\chi ]| \, {\mathrm {d}{\mathcal {H}}}^1 \leqq C \int _{J_\chi } |[\chi ]|^3 \, {\mathrm {d}{\mathcal {H}}}^1 = C H(\chi , \Omega ) < + \infty . \end{aligned}$$

In conclusion, $\mathrm {D}\chi = \mathrm {D}^j \chi \in \mathcal {M}_b(\Omega )$ and this concludes the proof. $\square$

5 Proof of Compactness

In this section we prove a series of results which lead to the compactness statement in Theorem 4.1-i). Some of the steps are inspired by the proof of compactness in the continuum setting in [26].

Proposition 5.1

Let $\Omega \subset \mathbb {R}^2$ be an open and bounded set. Let $(\chi _n)_n \in L^1_{\mathrm {loc}}(\mathbb {R}^2;\mathbb {R}^2)$ and $\chi \in L^1_{\mathrm {loc}}(\mathbb {R}^2;\mathbb {R}^2)$ be such that $\chi _n \rightarrow \chi$ in $L^1_{\mathrm {loc}}(\Omega ;\mathbb {R}^2)$ and

$$\begin{aligned} \sup _n H_n (\chi _n,\Omega ) < +\infty . \end{aligned}$$

Then $\chi$ solves

$$\begin{aligned} |\chi | = 1 \text { a.e. in } \Omega , \quad \mathrm {curl}(\chi ) = 0 \text { in } \mathcal {D}'(\Omega ). \end{aligned}$$

(5.1)

Proposition 5.2

Let $\Omega \subset \mathbb {R}^2$ be an open and bounded set. Let $(\chi _n)_n \in L^1_{\mathrm {loc}}(\mathbb {R}^2;\mathbb {R}^2)$ be such that

$$\begin{aligned} \sup _n H_n (\chi _n, \Omega ) < +\infty . \end{aligned}$$

Then there exists $\chi \in L^\infty (\mathbb {R}^2;\mathbb {S}^1)$ such that, up to a subsequence, $\chi _n \rightarrow \chi$ in $L^p_{\mathrm {loc}}(\Omega ;\mathbb {R}^2)$ for every $p \in [1,6)$.

Propositions  5.1 and 5.2 yield Theorem 4.1-i).

Proof of Proposition 5.1

By Proposition 2.6 we have that $\int _{\Omega '} W(\chi _n) \, {\mathrm {d}{x}} \leqq C \varepsilon _n$ for every $\Omega ' \subset \subset \Omega$ and as a consequence $|\chi _n|^2 \rightarrow 1$ in $L^2_{\mathrm {loc}}(\Omega )$. Thus, we find a (non-relabeled) subsequence with $|\chi _n| \rightarrow 1$ and $\chi _n \rightarrow \chi$ a.e. in $\Omega$. In particular, $|\chi | = 1$ a.e. in $\Omega$. To show that $\mathrm {curl}(\chi ) = 0$ in the distributional sense, let us recall that by Remark 2.5, $\mathrm {curl}^{\mathrm {d}}(\chi _n) \rightharpoonup 0$ in the sense of distributions. Thus it is sufficient to show that $\mathrm {curl}(\chi _n) - \mathrm {curl}^{\mathrm {d}}(\chi _n) \rightharpoonup 0$ in the sense of distributions. Using the interpolation ${\mathcal {I}}$ defined in (2.4), we have that $\mathrm {curl}(\chi _n) - \mathrm {curl}^{\mathrm {d}}(\chi _n) = - \mathrm {div}(\chi _n^\perp - {\mathcal {I}} (\chi _n^\perp ))$ as distributions. Moreover, using (2.5) and Proposition 2.6, we obtain that

$$\begin{aligned} \Vert \chi _n^\perp - {\mathcal {I}} (\chi _n^\perp ) \Vert _{L^2(\Omega ')} \leqq C \lambda _n\Vert \mathrm {D}^{\mathrm {d}}\chi _n \Vert _{L^2(\Omega ')} \leqq C \frac{\lambda _n}{\sqrt{\varepsilon _n}} = C \lambda _n^{1/2} \delta _n^{1/4} \rightarrow 0 \end{aligned}$$

for every $\Omega ' \subset \subset \Omega$, and the desired distributional convergence $\mathrm {curl}(\chi _n) - \mathrm {curl}^{\mathrm {d}}(\chi _n) \rightharpoonup 0$ follows.⁴$\square$

Proof of Proposition 5.2

Step 1. (Recasting the discrete entropy productions.) Let $\Phi \in \mathrm {Ent}$ and let $\alpha$ and $\Psi$ be as in (3.2) and (3.3). We show that there are discrete functions $r_n^{(1)}, r_n^{(2)} \in \mathcal {PC}_{\lambda _n}(\mathbb {R})$ such that

$$\begin{aligned} \mathrm {div}^{\mathrm {d}}(\Phi \circ \chi _n^\perp ) = (\Psi \circ \chi _n^\perp ) \cdot \mathrm {D}^{\mathrm {d}}(1 - |\chi _n|^2) + r_n^{(1)} + r_n^{(2)}, \end{aligned}$$

where $r_n^{(1)}$ and $r_n^{(2)}$ are estimated below in Step 5. By a discrete chain rule we get that

$$\begin{aligned} \mathrm {div}^{\mathrm {d}}( \Phi \circ \chi _n^\perp ) = \nabla \Phi _1(X_n) \cdot \partial ^{\mathrm {d}}_1 \chi _n^\perp + \nabla \Phi _2(Y_n) \cdot \partial ^{\mathrm {d}}_2 \chi _n^\perp \quad \text {in } \mathcal {PC}_{\lambda _n}(\mathbb {R}), \end{aligned}$$

where $(X_n)^{i,j}$ is a vector on the segment connecting $(\chi _n^\perp )^{i,j}$ and $(\chi _n^\perp )^{i+1,j}$, and $(Y_n)^{i,j}$ lies on the segment connecting $(\chi _n^\perp )^{i,j}$ and $(\chi _n^\perp )^{i,j+1}$. By (3.4) we get that

$$\begin{aligned} \begin{aligned} \mathrm {div}^{\mathrm {d}}( \Phi \circ \chi _n^\perp \big )&= \big ( \nabla \Phi _1(X_n) - \nabla \Phi _1(\chi _n^\perp ) \big ) \cdot \partial ^{\mathrm {d}}_1 \chi _n^\perp \\&\quad + \big ( \nabla \Phi _2(Y_n) - \nabla \Phi _2(\chi _n^\perp ) \big ) \cdot \partial ^{\mathrm {d}}_2 \chi _n^\perp \\&\quad -2 \chi _n^\perp \cdot \big ( \Psi _1( \chi _n^\perp ) \partial ^{\mathrm {d}}_1 \chi _n^\perp + \Psi _2( \chi _n^\perp ) \partial ^{\mathrm {d}}_2 \chi _n^\perp \big )\\&\quad + \alpha (\chi _n^\perp ) \big ( \partial ^{\mathrm {d}}_1 \chi _{1,n}^\perp + \partial ^{\mathrm {d}}_2 \chi _{2,n}^\perp \big ). \end{aligned} \end{aligned}$$

By a discrete chain rule we also have

$$\begin{aligned}&\partial ^{\mathrm {d}}_1(1 - |\chi _n|^2) = \partial ^{\mathrm {d}}_1 (1 - |\chi _n^\perp |^2) = -2 {\widetilde{X}}_n \cdot \partial ^{\mathrm {d}}_1 \chi _n^\perp \quad \text {and} \quad \\&\partial ^{\mathrm {d}}_2(1 - |\chi _n|^2) = -2 {\widetilde{Y}}_n \cdot \partial ^{\mathrm {d}}_2 \chi _n^\perp \, , \end{aligned}$$

where $({\widetilde{X}}_n)^{i,j}$ lies between $(\chi _n^\perp )^{i,j}$ and $(\chi _n^\perp )^{i+1,j}$, and $({\widetilde{Y}}_n)^{i,j}$ lies between $(\chi _n^\perp )^{i,j}$ and $(\chi _n^\perp )^{i,j+1}$. Therefore we get

$$\begin{aligned} \mathrm {div}^{\mathrm {d}}( \Phi \circ \chi _n^\perp \big ) = (\Psi \circ \chi _n^\perp ) \cdot \mathrm {D}^{\mathrm {d}}(1 - |\chi _n|^2) + r_n^{(1)} + r_n^{(2)}, \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} r_n^{(1)}&= \big ( \nabla \Phi _1(X_n) - \nabla \Phi _1(\chi _n^\perp ) \big ) \cdot \partial ^{\mathrm {d}}_1 \chi _n^\perp + \big ( \nabla \Phi _2(Y_n) - \nabla \Phi _2(\chi _n^\perp ) \big ) \cdot \partial ^{\mathrm {d}}_2 \chi _n^\perp \\&\quad -2 \Psi _1(\chi _n^\perp ) \, (\chi _n^\perp - {\widetilde{X}}_n) \cdot \partial ^{\mathrm {d}}_1 \chi _n^\perp -2 \Psi _2(\chi _n^\perp ) \, (\chi _n^\perp - {\widetilde{Y}}_n) \cdot \partial ^{\mathrm {d}}_2 \chi _n^\perp \\&\quad + \alpha (\chi _n^\perp ) \, \mathrm {div}^{\mathrm {d}}(\overline{\chi }_n^\perp ) \end{aligned} \end{aligned}$$

(5.2)

and

$$\begin{aligned} r_n^{(2)} = \alpha (\chi _n^\perp ) \, \mathrm {div}^{\mathrm {d}}( \chi _n^\perp - \overline{\chi }_n^\perp ). \end{aligned}$$

(5.3)

Here we recall that $\overline{\chi }_n$ is the linearized version of the order parameter $\chi _n$ defined by (2.19).

Step 2. (Estimates for the remainders $r_n^{(1)}$ and $r_n^{(2)}$.) By the Lipschitz continuity of $\mathrm {D}\Phi$ and the fact that $|(X_n)^{i,j} - (\chi _n^\perp )^{i,j}| \leqq |(\chi _n^\perp )^{i+1,j} - (\chi _n^\perp )^{i,j}|$ we have that

$$\begin{aligned} \big | \nabla \Phi _1(X_n) - \nabla \Phi _1(\chi _n^\perp ) \big | \leqq C |X_n - \chi _n^\perp | \leqq C \lambda _n|\partial ^{\mathrm {d}}_1 \chi _n^\perp |, \end{aligned}$$

(5.4)

a similar estimate being true for $\big | \nabla \Phi _2(Y_n) - \nabla \Phi _2(\chi _n^\perp ) \big |$. Similarly, by the boundedness of $\Psi$, we get that

$$\begin{aligned}&\big | \Psi _1(\chi _n^\perp ) \, (\chi _n^\perp - {\widetilde{X}}_n) \big | \leqq C \lambda _n|\partial ^{\mathrm {d}}_1 \chi _n^\perp | \nonumber \\&\quad \text {and} \quad \big | \Psi _2(\chi _n^\perp ) \, (\chi _n^\perp - {\widetilde{Y}}_n) \big | \leqq C \lambda _n|\partial ^{\mathrm {d}}_2 \chi _n^\perp |. \end{aligned}$$

(5.5)

Using (5.4) and (5.5) in (5.2), we get that

$$\begin{aligned} |r_n^{(1)}| \leqq C \lambda _n|\mathrm {D}^{\mathrm {d}}\chi _n|^2 + C |\mathrm {curl}^{\mathrm {d}}(\overline{\chi }_n)|, \end{aligned}$$

where we have also used the boundedness of $\alpha$ and the identity $|\mathrm {div}^{\mathrm {d}}(\overline{\chi }_n^\perp )| = |\mathrm {curl}^{\mathrm {d}}(\overline{\chi }_n)|$. For every $\Omega ' \subset \subset \Omega$ we have by Proposition 2.6 and (2.13) that $\Vert \lambda _n|\mathrm {D}^{\mathrm {d}}\chi _n|^2 \Vert _{L^1(\Omega ')} \leqq C \frac{\lambda _n}{\varepsilon _n} = C \sqrt{\delta _n}$ and by Lemma 2.4 that $\Vert \mathrm {curl}^{\mathrm {d}}(\overline{\chi }_n) \Vert _{L^1(\Omega ')} \leqq C \delta _n$. Therefore,

$$\begin{aligned} \Vert r_n^{(1)} \Vert _{L^1(\Omega ')} = \mathcal {O}(\sqrt{\delta _n}). \end{aligned}$$

Let us prove that

$$\begin{aligned} r_n^{(2)} \rightarrow 0 \quad \text {in } H^{-1}(\Omega ') \text { for every } \Omega ' \subset \subset \Omega . \end{aligned}$$

We first use boundedness of $\alpha$ to infer that $|r_n^{(2)}| \leqq C | \mathrm {div}^{\mathrm {d}}(\chi _n^\perp - \overline{\chi }_n^\perp )|$. As observed in Remark 2.5, the fact that $\chi _n^\perp - \overline{\chi }_n^\perp \rightarrow 0$ in $L^2(\Omega )$ implies that $\mathrm {div}^{\mathrm {d}}(\chi _n^\perp - \overline{\chi }_n^\perp ) \rightarrow 0$ in $H^{-1}(\Omega ')$ for every open $\Omega ' \subset \subset \Omega$ through the use of the interpolation $\mathcal {I}$ defined by (2.4). As a consequence, $r_n^{(2)} \rightarrow 0$ in $H^{-1}(\Omega ')$ for every $\Omega ' \subset \subset \Omega$ as desired.

Step 3. (Compactness in $H^{-1}$ of the discrete entropy productions.) Let us prove that the sequence $(\mathrm {div}^{\mathrm {d}}(\Phi \circ \chi _n^\perp ))_n$ is compact in $H^{-1}(\Omega ')$, for every $\Omega ' \subset \subset \Omega$. To this end we apply Lemma 5.4 below. Let us first show how to write $\mathrm {div}^{\mathrm {d}}(\Phi \circ \chi _n^\perp )$ as the distributional divergences of $L^2$ vector fields whose squares are uniformly integrable on $\Omega '$, where $\Omega ' \subset \subset \Omega$ is a fixed open set. Using again the interpolation ${\mathcal {I}}$ defined by (2.4), we get that $\mathrm {div}^{\mathrm {d}}(\Phi \circ \chi _n^\perp ) = \mathrm {div}(\mathcal {I} (\Phi \circ \chi _n^\perp ))$. Moreover, we observe that $(\mathcal {I} (\Phi \circ \chi _n^\perp ))_n$ is bounded in $L^\infty$ since $\Phi$ is a bounded function. As a consequence, $| \mathcal {I} (\Phi \circ \chi _n^\perp ) |^2$ is uniformly integrable on $\Omega '$.

To apply Lemmsa 5.4, let us now use a discrete product rule to write

$$\begin{aligned} \mathrm {div}^{\mathrm {d}}\big ( (\Psi \circ \chi _n^\perp ) (1 - |\chi _n|^2) \big ) = (\Psi \circ \chi _n^\perp ) \cdot \mathrm {D}^{\mathrm {d}}(1 - |\chi _n|^2) + R_n \quad \text {in } \mathcal {PC}_{\lambda _n}(\mathbb {R}), \end{aligned}$$

where

$$\begin{aligned} R_n^{i,j} = \partial ^{\mathrm {d}}_1 (\Psi _1 \circ \chi _n^\perp )^{i,j} (1 - |\chi _n|^2)^{i+1,j} + \partial ^{\mathrm {d}}_2 (\Psi _2 \circ \chi _n^\perp )^{i,j} (1 - |\chi _n|^2)^{i,j+1}. \end{aligned}$$

In view of Step 5 this leads to

$$\begin{aligned} \mathrm {div}^{\mathrm {d}}( \Phi \circ \chi _n^\perp ) = \mathrm {div}^{\mathrm {d}}\big ( (\Psi \circ \chi _n^\perp ) (1 - |\chi _n|^2) \big ) - R_n + r_n^{(1)} + r_n^{(2)} \end{aligned}$$

and we will show that

$$\begin{aligned}&(\textit{a}) \quad \mathrm {div}^{\mathrm {d}}\big ( (\Psi \circ \chi _n^\perp ) (1 - |\chi _n|^2) \big ) + r_n^{(2)} \rightarrow 0 \text { in } H^{-1}(\Omega ') \text { and} \nonumber \\&(\textit{b}) \quad - R_n + r_n^{(1)} \in L^2(\Omega '), \ \sup _n \Vert - R_n + r_n^{(1)} \Vert _{L^1(\Omega ')} < + \infty . \end{aligned}$$

(5.6)

By Step 5, to prove (a) in (5.6) it remains to show that $\mathrm {div}^{\mathrm {d}}\big ( (\Psi \circ \chi _n^\perp ) (1 - |\chi _n|^2) \big ) \rightarrow 0$ in $H^{-1}(\Omega ')$. Since $\Psi$ is a bounded function and $1 - |\chi _n|^2 \rightarrow 0$ in $L^2(\Omega )$ in view of Proposition 2.6, we can proceed as in the estimate of $r_n^{(2)}$ in Step 5: For the interpolated fields $\mathcal {I}\big ( (\Psi \circ \chi _n^\perp ) (1 - |\chi _n|^2) \big )$ defined by (2.4) we get that $\mathcal {I}\big ( (\Psi \circ \chi _n^\perp ) (1 - |\chi _n|^2) \big ) \rightarrow 0$ in $L^2(\Omega ')$ and $\mathrm {div}^{\mathrm {d}}\big ( (\Psi \circ \chi _n^\perp ) (1 - |\chi _n|^2) \big ) = \mathrm {div}\big ( \mathcal {I} \big ( (\Psi \circ \chi _n^\perp ) (1 - |\chi _n|^2) \big ) \big )$. Thereby, the desired convergence to 0 in $H^{-1}(\Omega ')$ follows.

To prove (b) in (5.6), we first observe that for every fixed n, $r_n^{(1)}$ and $R_n$ belong to $L^\infty (\Omega ')$ since they only attain finitely many values on $\Omega '$. In view of Step 5 it remains only to show that $(R_n)_n$ is bounded in $L^1(\Omega ')$. We observe that $|\mathrm {D}^{\mathrm {d}}(\Psi \circ \chi _n^\perp )| \leqq C |\mathrm {D}^{\mathrm {d}}\chi _n|$ since $\Psi$ is a Lipschitz function. By Young’s inequality we get that

$$\begin{aligned} |R_n^{i,j}| \leqq C \Big ( \varepsilon _n|\mathrm {D}^{\mathrm {d}}\chi _n^{i,j}|^2 + \frac{1}{\varepsilon _n} \big ( (1 - |\chi _n^{i+1,j}|^2 )^2 + (1 - |\chi _n^{i,j+1}|^2 )^2 \big ) \Big ), \end{aligned}$$

and we obtain boundedness in $L^1(\Omega ')$ from Proposition 2.6.

Step 4. (Compactness in $H^{-1}$ of the distributional entropy productions.)

Let us prove that the sequence $(\mathrm {div}(\Phi \circ \chi _n^\perp ))_n$ is compact in $H^{-1}(\Omega ')$, for every $\Omega ' \subset \subset \Omega$.

We again use the interpolation defined by (2.4): for every $\Omega ' \subset \subset \Omega$, $\mathrm {div}( \mathcal {I} (\Phi \circ \chi _n^\perp )) = \mathrm {div}^{\mathrm {d}}(\Phi \circ \chi _n^\perp )$ is compact in $H^{-1}(\Omega ')$ by Step 5 and, as a consequence, it is enough to show that $\mathcal {I} (\Phi \circ \chi _n^\perp ) - (\Phi \circ \chi _n^\perp ) \rightarrow 0$ in $L^2(\Omega ')$. Using the Lipschitz continuity of $\Phi$ we have that $|\mathcal {I} (\Phi \circ \chi _n^\perp ) - (\Phi \circ \chi _n^\perp ) | \leqq C \lambda _n|\mathrm {D}^{\mathrm {d}}(\Phi \circ \chi _n^\perp )| \leqq C \lambda _n|\mathrm {D}^{\mathrm {d}}\chi _n|$ and in view of Proposition 2.6 and (2.13) this yields that $\Vert \mathcal {I} (\Phi \circ \chi _n^\perp ) - (\Phi \circ \chi _n^\perp ) \Vert _{L^2(\Omega ')} \leqq C \frac{\lambda _n}{\sqrt{\varepsilon _n}} \rightarrow 0$.

Step 5. (Bounds in $L^6$ for $\chi _n$.) By Proposition 2.7 the sequence $(\chi _n)_n$ is bounded in $L^6(\Omega ')$ for every $\Omega ' \subset \subset \Omega$.

Step 6. (Compactness in $L^p_{\mathrm {loc}}$, $p \in [1,6)$, for $\chi _n$.) We fix again $\Omega ' \subset \subset \Omega$. We will show that there exists a $\chi \in L^\infty (\Omega '; \mathbb {S}^1)$ and a (non-relabeled) subsequence $\chi _n \rightarrow \chi$ in $L^p(\Omega '; \mathbb {R}^2)$ for all $p \in [1,6)$. The claim of Proposition 5.2 then finally follows by exhausting $\Omega$ with a sequence of compactly contained subsets and using a diagonal argument. To prove the compactness in $L^p(\Omega ')$, $p < 6$, we make use of the theory of Young measures. There exists a (non-relabeled) subsequence of $(\chi _n^\perp )_n$ and a Young measure $\nu = (\nu _x)_{x \in \Omega '}$ such that for every $g \in C_0(\mathbb {R}^2)$ we have that

$$\begin{aligned} g \circ \chi _n^\perp {\mathop {\rightharpoonup }\limits ^{*}}\overline{g} \text { weakly* in } L^\infty (\Omega '), \text { where } \overline{g}(x) = \int _{\mathbb {R}^2} g \, {\mathrm {d}{\nu }}_x. \end{aligned}$$

(5.7)

For later use, let us record several additional properties of the Young measure $\nu$: By Proposition 2.6 we have that $\int _{\Omega '} (1 - |\chi _n^\perp |^2)^2 \, {\mathrm {d}{x}} \leqq C \varepsilon _n\rightarrow 0$ and, as a consequence, $\nu _x$ is supported on $\mathbb {S}^1$ for a.e. $x \in \Omega '$.⁵ Since $(\chi _n^\perp )_n$ is bounded in $L^6(\Omega ')$ by Step 5, we moreover have that $\nu _x$ is a probability measure for a.e. $x \in \Omega '$⁶ and that

$$\begin{aligned} g \circ \chi _n^\perp \rightharpoonup \overline{g} \text { weakly in } L^{6/p}(\Omega ') \, , \text { where } \overline{g}(x) = \int _{\mathbb {R}^2} g \, {\mathrm {d}{\nu }}_x \end{aligned}$$

(5.8)

for every $p < 6$ and every function $g \in C(\mathbb {R}^2)$ with $|g(\xi )| \leqq C(1 + |\xi |^p)$.⁷ In particular, taking as g the components of the identity on $\mathbb {R}^2$ we get that $(\chi _n^\perp )_n$ itself converges weakly in $L^6(\Omega ')$.

To improve this to strong convergence, we will now show that for a.e. $x \in \Omega '$, $\nu _x$ is a Dirac measure. For the moment, let us fix two entropies $\Phi _{(1)}, \Phi _{(2)} \in \mathrm {Ent}$. Applying (5.7) to the components of $\Phi _{(1)}$ and $\Phi _{(2)}$ we get that

$$\begin{aligned} \Phi _{(k)} \circ \chi _n^\perp {\mathop {\rightharpoonup }\limits ^{*}}\overline{\Phi }_{(k)} \text { weakly* in } L^\infty (\Omega '; \mathbb {R}^2), \quad \overline{\Phi }_{(k)}(x) = \int _{\mathbb {R}^2} \Phi _{(k)} \, {\mathrm {d}{\nu }}_x \end{aligned}$$

for $k = 1,2$. Now we recall that by Step 5, $\big ( \mathrm {div}(\Phi _{(1)} \circ \chi _n^\perp ) \big )_n$ and $\big ( \mathrm {curl}(\Phi _{(2)}^\perp \circ \chi _n^\perp ) \big )_n = \big ( \mathrm {div}(\Phi _{(2)} \circ \chi _n^\perp ) \big )_n$ are compact in $H^{-1}(\Omega ')$. Therefore, the div-curl lemma (cf. [44, 54]) yields that

$$\begin{aligned} (\Phi _{(1)} \circ \chi _n^\perp ) \cdot (\Phi _{(2)}^\perp \circ \chi _n^\perp ) \rightharpoonup \overline{\Phi }_{(1)} \cdot \overline{\Phi }_{(2)}^\perp \quad \text {in the sense of distributions on } \Omega '. \end{aligned}$$

On the other hand, (5.7) applied to $\Phi _{(1)} \cdot \Phi _{(2)}^\perp$ leads to

$$\begin{aligned} (\Phi _{(1)} \circ \chi _n^\perp ) \cdot (\Phi _{(2)}^\perp \circ \chi _n^\perp ) {\mathop {\rightharpoonup }\limits ^{*}}\overline{ \Phi _{(1)} \cdot \Phi _{(2)}^\perp } \text { weakly* in } L^\infty (\Omega '; \mathbb {R}^2). \end{aligned}$$

In conclusion,

$$\begin{aligned} \bigg ( \int _{\mathbb {R}^2} \Phi _{(1)} \, {\mathrm {d}{\nu }}_x \bigg ) \cdot \bigg ( \int _{\mathbb {R}^2} \Phi _{(2)}^\perp \, {\mathrm {d}{\nu }}_x \bigg ) = \bigg ( \int _{\mathbb {R}^2} \Phi _{(1)} \cdot \Phi _{(2)}^\perp \, {\mathrm {d}{\nu }}_x \bigg ) \quad \text {for a.e.\ } x \in \Omega '. \end{aligned}$$

The exceptional null set depends on $\Phi _{(1)}, \Phi _{(2)} \in \mathrm {Ent}$. Nonetheless, we can get rid of this dependence since both sides of the above equation are continuous under uniform convergence of $\Phi _{(1)}, \Phi _{(2)}$ and since the space $\mathrm {Ent}$ is separable with respect to the $L^\infty$ norm, being a subspace of the separable metric space $C_0(\mathbb {R}^2;\mathbb {R}^2)$. This allows us to apply [26, Lemma 2.6] to obtain that $\nu _x$ is a Dirac measure for a.e. $x \in \Omega '$.⁸ To this end let us recall that we have already shown that $\nu _x$ is supported on $\mathbb {S}^1$ for a.e. x.

Defining

$$\begin{aligned} \chi (x) := \int _{\mathbb {R}^2} -\xi ^\perp \, {\mathrm {d}{\nu }}_x(\xi ), \quad x \in \Omega ', \end{aligned}$$

we now have that $\chi \in L^\infty (\mathbb {R}^2; \mathbb {S}^1)$ and for a.e. $x \in \Omega '$, $\nu _x$ is the Dirac measure in the point $\chi ^\perp (x)$. Applying (5.8) with $p=1$ to the components of $\xi \mapsto - \xi ^\perp$ we moreover obtain that $\chi _n \rightharpoonup \chi$ weakly in $L^6(\Omega ')$. Now let us fix $p \in [1,6)$ and show that the convergence is in fact strong in $L^p(\Omega ')$. Applying (5.8) to $g(\xi ) = |\xi |^p$ we get that $|\chi _n|^p \rightharpoonup |\chi |^p$ weakly in $L^{6/p}(\Omega ')$ because $\nu _x$ is the Dirac measure in the point $\chi ^\perp (x)$. Testing this weak convergence with the characteristic function of $\Omega '$ we get that $\Vert \chi _n \Vert _{L^p(\Omega ')}^p \rightarrow \Vert \chi \Vert _{L^p(\Omega ')}^p$. Since convergence of the norms improves weak convergence to strong convergence in $L^q$ for $q > 1$, we conclude that $\chi _n \rightarrow \chi$ strongly in $L^p(\Omega ')$. This concludes the proof of Proposition 5.2. $\square$

Remark 5.3

The same strategy can be used to prove the following compactness result for the discrete Aviles–Giga functionals $AG^{\mathrm {d}}_n$ defined by (2.23): If $AG^{\mathrm {d}}_n(\varphi _n,\Omega ) \leqq C$, then, up to a subsequence, $\mathrm {D}^{\mathrm {d}}\varphi _n$ converges in $L^p_{\mathrm {loc}}(\Omega )$ for every $p < 6$ and the limit is curl-free and valued in $\mathbb {S}^1$ a.e.

In fact, several of the steps in the proofs of Propositions 5.1 and 5.2 simplify due to the fact that when $\chi _n = \mathrm {D}^{\mathrm {d}}\varphi _n$, we have that $\mathrm {curl}^{\mathrm {d}}(\chi _n) \equiv 0$ in place of only $\mathrm {curl}^{\mathrm {d}}(\chi _n) \simeq 0$. In particular, the term $\alpha (\chi _n^\perp ) \mathrm {div}^{\mathrm {d}}(\overline{\chi }_n^\perp )$ in (5.2) as well as the remainder $r_n^{(2)}$ in (5.3) are not present. Then, all later steps in the proof of Proposition 5.2 apply with only few obvious modifications, noting that the bounds obtained applying Proposition 2.6 follow in this case directly from the energy bound $AG^{\mathrm {d}}_n(\varphi _n,\Omega ) \leqq C$.

We conclude this section by stating and proving a technical result used in the proof of Proposition 5.2. It is a slightly modified version of [26, Lemma 3.1]. Nevertheless, we provide the proof for completeness.

Lemma 5.4

Let $U \subset \mathbb {R}^d$ be an open bounded set. Let $(f_n)_n$ be a sequence in $L^2(U;\mathbb {R}^d)$ such that $(|f_n|^2)_n$ is uniformly integrable. If $\mathrm {div}(f_n) = a_n + b_n$, where $(a_n)_n$ is compact in $H^{-1}(U)$ and $(b_n)_n$ is a sequence in $L^2(U)$ with $\sup _n \Vert b_n\Vert _{L^1(U)} < +\infty$, then $(\mathrm {div}(f_n))_n$ is compact in $H^{-1}(U)$.

Proof

Let us fix a sequence $(\varphi _n)_n$ in $H^1_0(U)$ such that $\varphi _n \rightharpoonup 0$ weakly in $H^1_0(U)$. We will prove that $\langle \mathrm {div}(f_n) , \varphi _n \rangle _{H^{-1}(U), H^1_0(U)} \rightarrow 0$.⁹ For such a sequence $(\varphi _n)$, we have that $\varphi _n \rightarrow 0$ strongly in $L^2(U)$, and, in particular, that

$$\begin{aligned} \mathcal {L}^d(U \cap \{ | \varphi _n |> \delta \} ) \rightarrow 0 \quad \text {for every } \delta > 0. \end{aligned}$$

(5.9)

We fix $\delta > 0$, and define the truncated functions

$$\begin{aligned} \varphi _n^{(1)} := {\left\{ \begin{array}{ll} - \delta &{} \text {on } \{ \varphi _n < - \delta \}, \\ \varphi _n &{} \text {on } \{ |\varphi _n | \leqq \delta \}, \\ \delta &{} \text {on } \{ \varphi _n > \delta \}, \end{array}\right. } \end{aligned}$$

and then have $\varphi _n^{(1)} \in H^1_0(U)$ with $\nabla \varphi _n^{(1)} = \nabla \varphi _n \cdot \mathbbm {1}_{\{ | \varphi _n | \leqq \delta \} }$. We moreover set $\varphi _n^{(2)} := \varphi _n - \varphi _n^{(1)}$. We claim that $\varphi _n^{(2)} \rightharpoonup 0$ in $H^1_0(U)$ and therefore also $\varphi _n^{(1)} \rightharpoonup 0$ in $H^1_0 (U)$. To prove this claim, let $\psi ^* \in H^{-1}(U)$. Let $\psi \in H^1_0(U)$ solve $-\Delta \psi = \psi ^*$. Then,

$$\begin{aligned} | \langle \psi ^*, \varphi _n^{(2)} \rangle | = \bigg | \int _{\{|\varphi _n |> \delta \}}{\nabla \psi \cdot \nabla \varphi _n }{\, {\mathrm {d}{x}}} \bigg | \leqq \Vert \nabla \psi \Vert _{L^2(\{| \varphi _n | > \delta \})} \Vert \nabla \varphi _n \Vert _{L^2(U)}. \end{aligned}$$

By (5.9) and since weak convergence of $\varphi _n$ in $H^1_0(U)$ implies that $\Vert \nabla \varphi _n \Vert _{L^2(U)}$ is bounded, we infer that $\langle \psi ^*, \varphi _n^{(2)} \rangle \rightarrow 0$, which proves our claim.

Now we write $\langle \mathrm {div}(f_n) , \varphi _n \rangle = \langle a_n , \varphi _n^{(1)} \rangle + \langle b_n , \varphi _n^{(1)} \rangle + \langle \mathrm {div}(f_n) , \varphi _n^{(2)} \rangle$. Since $(a_n)_n$ is compact in $H^{-1}(U)$, we have that $\langle a_n, \varphi _n^{(1)} \rangle \rightarrow 0$. Moreover, since $b_n$ are functions in $L^2(U)$, the $\big (H^{-1}(U), H^1_0(U)\big )$-pairing between $b_n$ and $\varphi _n^{(1)}$ is given by $\int _U b_n \, \varphi _n^{(1)} \, {\mathrm {d}{x}}$ and thus we have that

$$\begin{aligned} | \langle b_n , \varphi _n^{(1)} \rangle | = \bigg | \int _{U}{ b_n \, \varphi _n^{(1)} }{\, {\mathrm {d}{x}}} \bigg | \leqq \delta \sup _{n} \Vert b_n \Vert _{L^1(U)}. \end{aligned}$$

Finally,

$$\begin{aligned} | \langle \mathrm {div}(f_n) , \varphi _n^{(2)} \rangle | = \bigg | \int _{U}{ f_n \cdot \nabla \varphi _n^{(2)} }{\, {\mathrm {d}{x}}} \bigg | \leqq \Vert f_n \Vert _{L^2(\{ |\varphi _n| > \delta \})} \Vert \varphi _n \Vert _{L^2(U)}, \end{aligned}$$

which goes to zero by boundedness of $(\varphi _n)_n$ in $L^2(U)$, by (5.9), and by the uniform integrability of $(|f_n|^2)_n$. In conclusion we obtain that

$$\begin{aligned} \limsup _{n \rightarrow \infty } | \langle \mathrm {div}(f_n) , \varphi _n \rangle | \leqq \delta \sup _{n} \Vert b_n \Vert _{L^1(U)}. \end{aligned}$$

Since $\delta > 0$ is arbitrary and $(b_n)_n$ is bounded in $L^1(U)$, this concludes the proof. $\square$

6 Proof of the Liminf Inequality

In this section we prove Theorem 4.1-ii). We assume for the whole section that $\Omega \subset \mathbb {R}^2$ is an open and bounded set. Let us fix $(\chi _n)_n$ and $\chi \in L^1_{\mathrm {loc}}(\mathbb {R}^2;\mathbb {R}^2)$ such that $\chi _n \rightarrow \chi$ in $L^1_{\mathrm {loc}}(\Omega ;\mathbb {R}^2)$. Let us assume, without loss of generality, that $\liminf _n H_n(\chi _n, \Omega ) = \lim _n H_n(\chi _n, \Omega ) < + \infty$. By Proposition 5.1 we get that $\chi$ satisfies (5.1), i.e., the first two conditions in (3.6). In the following we prove (4.2), which yields, in particular, the third condition in (3.6).

Let us fix $\Phi \in \mathrm {Ent}$ with $\Vert \Phi \Vert _{\mathrm {Ent}} \leqq 1$. We let $\Psi$ and $\alpha$ denote the functions given by (3.2), (3.3). We start by noticing that the condition $|\chi | = 1$ a.e. in $\Omega$ yields

$$\begin{aligned} \Phi \circ \chi ^\perp = {\widetilde{\Phi }} \circ \chi ^\perp \quad \text {a.e. in } \Omega , \end{aligned}$$

where ${\widetilde{\Phi }}(\xi ) := \Phi (\xi ) - (1 - |\xi |^2) \Psi (\xi )$. Hence, it suffices to estimate the total variation of $\mathrm {div}({\widetilde{\Phi }} \circ \chi ^\perp )$.

Remark 6.1

(Heuristic argument in a continuum setting) We estimate the total variation of $\mathrm {div}({\widetilde{\Phi }} \circ \chi ^\perp )$ below in several steps. To outline the proof, we first illustrate the argument in a continuum setting. Assume that $\omega _n \in H^1(\Omega ;\mathbb {R}^2)$, $\mathrm {curl}(\omega _n) = 0$, $\omega _n \rightarrow \chi$ in $L^1_{\mathrm {loc}}(\mathbb {R}^2;\mathbb {R}^2)$ and $\sup _n \frac{1}{2} \int _\Omega \frac{1}{\varepsilon _n} W(\omega _n) + \varepsilon _n|\mathrm {div}(\omega _n)|^2 \, {\mathrm {d}{x}}<\infty$. In the following we sketch how to show that

$$\begin{aligned} |\mathrm {div}({\widetilde{\Phi }} \circ \chi ^\perp )|(\Omega ) \leqq \liminf _n \frac{1}{2} \int _\Omega \frac{1}{\varepsilon _n} W(\omega _n) + \varepsilon _n|\mathrm {div}(\omega _n)|^2 \, {\mathrm {d}{x}}. \end{aligned}$$

(6.1)

Note that the energies on the right-hand side of (6.1) are continuum analogues of our energies $H_n$. In view of Proposition 2.6, let us assume moreover that

$$\begin{aligned} \sup _n \frac{1}{2} \int _\Omega \frac{1}{\varepsilon _n} (1-|\omega _n|^2)^2 + \varepsilon _n|\mathrm {D}\omega _n|^2 \, {\mathrm {d}{x}} < + \infty . \end{aligned}$$

(6.2)

Step (Passing to the limit.) Given $\zeta \in C^\infty _c(\Omega )$ we have that

$$\begin{aligned} \langle \mathrm {div}({\widetilde{\Phi }} \circ \chi ^\perp ), \zeta \rangle = \lim _n \int _\Omega \zeta \mathrm {div}({\widetilde{\Phi }} \circ \omega _n^\perp ) \, {\mathrm {d}{x}}. \end{aligned}$$

Step (Expanding the divergence using (3.4).) The relation (3.4) yields that

$$\begin{aligned}&\mathrm {div}({\widetilde{\Phi }} \circ \omega _n^\perp ) = \alpha (\omega _n^\perp ) \mathrm {div}(\omega _n^\perp ) - q(\omega _n) \mathrm {div}(\Psi \circ \omega _n^\perp )\nonumber \\&\quad = - q(\omega _n) \mathrm {div}(\Psi \circ \omega _n^\perp ), \end{aligned}$$

(6.3)

where we have put $q(\xi ) := (1-|\xi |^2)$ and used that $\mathrm {curl}(\omega _n) = 0$.

Step (Young’s inequality.) By Young’s inequality we have that

$$\begin{aligned} \begin{aligned} - \int _\Omega \zeta q(\omega _n) \mathrm {div}(\Psi \circ \omega _n^\perp ) \, {\mathrm {d}{x}}&\leqq \frac{1}{2} \int _\Omega \frac{1}{\varepsilon _n} q(\omega _n)^2 + \varepsilon _n|\zeta |^2 |\mathrm {div}(\Psi \circ \omega _n^\perp )|^2 \, {\mathrm {d}{x}} \\&= \frac{1}{2} \int _\Omega \frac{1}{\varepsilon _n} W(\omega _n) + \varepsilon _n|\zeta |^2 |\mathrm {div}(\Psi \circ \omega _n^\perp )|^2 \, {\mathrm {d}{x}}, \end{aligned} \end{aligned}$$

where we have used that $q(\xi )^2 = W(\xi )$.

Step (From divergence to full derivative matrix.) We have that

$$\begin{aligned} \begin{aligned}&\varepsilon _n\int _\Omega |\zeta |^2 |\mathrm {D}(\Psi \circ \omega _n^\perp )|^2 \, {\mathrm {d}{x}} \\&\quad = \varepsilon _n\int _\Omega |\zeta |^2 \Big ( |\mathrm {div}(\Psi \circ \omega _n^\perp ) |^2 + |\mathrm {curl}(\Psi \circ \omega _n^\perp ) |^2 \\&\qquad - 2 \det \mathrm {D}(\Psi \circ \omega _n^\perp ) \Big ) \, {\mathrm {d}{x}} \\&\quad \geqq \varepsilon _n\int _\Omega |\zeta |^2 |\mathrm {div}(\Psi \circ \omega _n^\perp ) |^2 \, {\mathrm {d}{x}} + o_n(1), \end{aligned} \end{aligned}$$

(6.4)

where we have used that $\det \mathrm {D}(\Psi \circ \omega _n^\perp ) = \mathrm {curl}\big ( (\Psi _1 \circ \omega _n^\perp ) \nabla (\Psi _2 \circ \omega _n^\perp ) \big )$ and thus, integrating by parts,

$$\begin{aligned} \begin{aligned} \Big | \int _\Omega |\zeta |^2 \det \mathrm {D}(\Psi \circ \omega _n^\perp ) \, {\mathrm {d}{x}} \Big |&= \Big | \int _\Omega \nabla ^\perp (|\zeta |^2) \cdot \big ((\Psi _1 \circ \omega _n^\perp ) \nabla (\Psi _2 \circ \omega _n^\perp )\big ) \, {\mathrm {d}{x}} \Big | \\&\leqq C \Vert \mathrm {D}(\Psi \circ \omega _n^\perp )\Vert _{L^2} \leqq C \Vert \mathrm {D}\omega _n \Vert _{L^2} \leqq \frac{C}{\sqrt{\varepsilon _n}}. \end{aligned} \end{aligned}$$

Here we have used (6.2) and the fact that $\mathrm {Lip}(\Psi ) = \Vert \Phi \Vert _{\mathrm {Ent}} \leqq 1$ implies that $|\mathrm {D}(\Psi \circ \omega _n^\perp )| \leqq |\mathrm {D}\omega _n|$. Using the latter in (6.4) we now obtain that

$$\begin{aligned} \varepsilon _n\int _\Omega |\zeta |^2 |\mathrm {div}(\Psi \circ \omega _n^\perp ) |^2 \, {\mathrm {d}{x}} \leqq \varepsilon _n\int _\Omega |\zeta |^2 |\mathrm {D}\omega _n|^2 \, {\mathrm {d}{x}} + o_n(1). \end{aligned}$$

Step (From full derivative matrix to divergence.) Similarly to the previous step we get that

$$\begin{aligned} \varepsilon _n\int _\Omega |\zeta |^2 |\mathrm {D}\omega _n|^2 \, {\mathrm {d}{x}} = \varepsilon _n\int _\Omega |\zeta |^2 |\mathrm {div}(\omega _n) |^2 \, {\mathrm {d}{x}} + o_n(1), \end{aligned}$$

where we have used that $\mathrm {curl}(\omega _n) = 0$ and

$$\begin{aligned} \Big | \int _\Omega |\zeta |^2 \det \mathrm {D}\omega _n \, {\mathrm {d}{x}} \Big |= & {} \Big | \int _\Omega \nabla ^\perp (|\zeta |^2) \cdot \big (\omega _{1,n} \nabla \omega _{2,n} \big ) \, {\mathrm {d}{x}} \Big |\\\leqq & {} C \Vert \omega _n \Vert _{L^2} \Vert \mathrm {D}\omega _n\Vert _{L^2} \leqq \frac{C}{\sqrt{\varepsilon _n}}. \end{aligned}$$

By all the previous steps we now get that

$$\begin{aligned} \langle \mathrm {div}({\widetilde{\Phi }} \circ \chi ^\perp ), \zeta \rangle \leqq \liminf _n \frac{1}{2} \int _\Omega \frac{1}{\varepsilon _n} W(\omega _n) + \varepsilon _n|\mathrm {div}(\omega _n)|^2 \, {\mathrm {d}{x}} \end{aligned}$$

and taking the supremum over $\zeta$ we obtain (6.1).

To follow the previous steps in the discrete setting, we first need to introduce functions $q_n$ such that $W(\chi _n) = q_n(\overline{\chi }_n)^2$, namely

$$\begin{aligned} q_n(\xi ) := 1 - \frac{4}{\delta _n} \sin ^2 \Big ( \frac{\sqrt{\delta _n}}{2} \xi _1 \Big ) - \frac{4}{\delta _n} \sin ^2 \Big ( \frac{\sqrt{\delta _n}}{2} \xi _2 \Big ). \end{aligned}$$

Here we recall that $\overline{\chi }_n$ is the linearized version of the order parameter $\chi _n$ defined by (2.19). The functions $q_n$ are approximations of the function q. In fact, as we observe in the proof of Lemma 6.3 below (cf. (6.36)), they converge locally in $C^k(\mathbb {R}^2)$ for every k. Moreover, we introduce suitable approximations ${\widetilde{\Phi }}_n \in C_c^\infty (\mathbb {R}^2 \setminus \{0\};\mathbb {R}^2)$ of ${\widetilde{\Phi }}$ and functions $\Psi _n \in C_c^\infty (\mathbb {R}^2 \setminus \{0\};\mathbb {R}^2)$ and $\alpha _n \in C_c^\infty (\mathbb {R}^2 \setminus \{0\})$ with the properties that

$$\begin{aligned}&{\widetilde{\Phi }}_n \rightarrow {\widetilde{\Phi }}, \ \Psi _n \rightarrow \Psi , \ \alpha _n \rightarrow \alpha \quad \text {in } C^2, \end{aligned}$$

(6.5)

$$\begin{aligned}&\mathrm {D}{\widetilde{\Phi }}_n = \alpha _n \mathrm {Id} - q_n \mathrm {D}\Psi _n \text { in } \mathbb {R}^2, \end{aligned}$$

(6.6)

$$\begin{aligned}&\mathrm {Lip}(\Psi _n) \rightarrow \mathrm {Lip}(\Psi ) = \Vert \Phi \Vert _{\mathrm {Ent}} \, , \end{aligned}$$

(6.7)

$$\begin{aligned}&\mathrm {supp}({\widetilde{\Phi }}_n), \ \mathrm {supp}(\Psi _n), \ \mathrm {supp}(\alpha _n) \subset \subset (-M,M)^2 \end{aligned}$$

(6.8)

for some $M > 1$ independent of n. The existence of the latter approximations is proved in Lemma 6.3 below.

The reason to make use of these approximations is that, by using (6.6), they allow us to prove a relation similar to (6.3), namely (in a formal fashion)

$$\begin{aligned} \mathrm {div}^{\mathrm {d}}({\widetilde{\Phi }}_n \circ \overline{\chi }_n^\perp ) \simeq - q_n(\overline{\chi }_n) \mathrm {div}^{\mathrm {d}}(\Psi _n \circ \overline{\chi }_n^\perp ). \end{aligned}$$

The precise relation is obtained in (6.21) below. As can be seen below in Step 6, the fact that $q_n$ appears in place of q in the above formula allows us to recover the potential term in the energy $H_n$.

In the next steps, let us fix an open set $\Omega ' \subset \Omega$ and $\zeta \in C^\infty _c(\Omega ')$ with $\Vert \zeta \Vert _{L^\infty (\Omega ')} \leqq 1$ and let us prove that

$$\begin{aligned} \langle \mathrm {div}({\widetilde{\Phi }} \circ \chi ^\perp ), \zeta \rangle \leqq \liminf _{n \rightarrow \infty } \frac{1}{2} \int _{\Omega '} \frac{1}{\varepsilon _n} W^\mathrm {d}(\chi _n) + \varepsilon _n|A^\mathrm {d}(\chi _n)|^2 \, {\mathrm {d}{x}}. \end{aligned}$$

(6.9)

Replacing $\Omega '$ by a sufficiently small neighborhood of $\mathrm {supp}(\zeta )$ if necessary, we may assume, without loss of generality, that $\Omega ' \subset \subset \Omega$.

Step 1. (Passing to the limit.) We prove that

$$\begin{aligned} \langle \mathrm {div}({\widetilde{\Phi }} \circ \chi ^\perp ), \zeta \rangle = \lim _n \int _{\Omega '}\zeta \mathrm {div}^{\mathrm {d}}( {\widetilde{\Phi }}_n \circ \overline{\chi }_n^\perp ) \, {\mathrm {d}{x}}. \end{aligned}$$

(6.10)

This follows from the fact that $\mathrm {div}^{\mathrm {d}}( {\widetilde{\Phi }}_n \circ \overline{\chi }_n^\perp ) \rightharpoonup \mathrm {div}({\widetilde{\Phi }} \circ \chi ^\perp )$ in the sense of distributions. Indeed, we have that ${\widetilde{\Phi }}_n \circ \overline{\chi }_n^\perp \rightarrow {\widetilde{\Phi }} \circ \chi ^\perp$ in $L^1_{\mathrm {loc}}(\Omega ;\mathbb {R}^2)$ and $\mathrm {div}^{\mathrm {d}}( {\widetilde{\Phi }}_n \circ \overline{\chi }_n^\perp ) - \mathrm {div}( {\widetilde{\Phi }}_n \circ \overline{\chi }_n^\perp ) \rightharpoonup 0$ in $\mathcal {D}'(\Omega )$. The former is a consequence of (6.5), our assumption that $\chi _n \rightarrow \chi$ in $L^1_{\mathrm {loc}}(\Omega ; \mathbb {R}^2)$, and the fact that $\chi _n - \overline{\chi }_n \rightarrow 0$ in $L^2(\Omega ;\mathbb {R}^2)$ (cf. Remark 2.5). On the other hand, the latter is proved by observing that $\mathrm {div}^{\mathrm {d}}( {\widetilde{\Phi }}_n \circ \overline{\chi }_n^\perp ) - \mathrm {div}( {\widetilde{\Phi }}_n \circ \overline{\chi }_n^\perp ) = \mathrm {div}\big ( \mathcal {I} ( {\widetilde{\Phi }}_n \circ \overline{\chi }_n^\perp ) - ( {\widetilde{\Phi }}_n \circ \overline{\chi }_n^\perp ) \big )$, where ${\mathcal {I}}$ is defined by (2.4), and that

$$\begin{aligned}&| \mathcal {I} ( {\widetilde{\Phi }}_n \circ \overline{\chi }_n^\perp ) - ( {\widetilde{\Phi }}_n \circ \overline{\chi }_n^\perp ) | \\&\quad \leqq C \lambda _n|\mathrm {D}^{\mathrm {d}}( {\widetilde{\Phi }}_n \circ \overline{\chi }_n^\perp )| \leqq C \lambda _n|\mathrm {D}^{\mathrm {d}}\overline{\chi }_n| \rightarrow 0 \quad \text { in } L^2_{\mathrm {loc}}(\Omega ). \end{aligned}$$

Here we have used the fact that ${\widetilde{\Phi }}_n$ are equi-Lipschitz and (2.24) in the proof of Proposition 2.6.¹⁰

Step 2. (Removing cells where $\overline{\chi }_n^\perp$ lies outside of the support of ${\widetilde{\Phi }}_n$.) In the integral in (6.10) we remove the cells where $\overline{\chi }_n$ is far from zero by exploiting that ${\widetilde{\Phi }}_n$ have compact support.¹¹ More precisely, we fix $M>1$ such that $\mathrm {supp}({\widetilde{\Phi }}_n) \subset \subset (-M,M)^2$ for all n (cf. (6.8)) and we introduce the collection of cells

$$\begin{aligned} \begin{aligned}&\mathcal {Q}^{\mathrm {supp}}_n := \big \{ Q_{\lambda _n}(i,j) \ : \ (i,j) \in \mathbb {Z}^2, \ Q_{\lambda _n}(i,j) \cap \Omega ' \ne \emptyset , \\&\quad |\overline{\chi }_n| \leqq M \text { on } Q_{\lambda _n}(i,j) \text { and on all of its adjacent cells} \big \}. \end{aligned} \end{aligned}$$

(6.11)

By “adjacent cells” we mean that they share a side. We claim that

$$\begin{aligned} \int _{\Omega '} \zeta \mathrm {div}^{\mathrm {d}}( {\widetilde{\Phi }}_n \circ \overline{\chi }_n^\perp ) \, {\mathrm {d}{x}} = \int _{\Omega _n^{\mathrm {supp}}} \zeta \mathrm {div}^{\mathrm {d}}( {\widetilde{\Phi }}_n \circ \overline{\chi }_n^\perp ) \, {\mathrm {d}{x}} + o_n(1), \end{aligned}$$

(6.12)

where $\Omega _n^{\mathrm {supp}} := \Omega ' \cap \bigcup _{Q \in \mathcal {Q}_n^{\mathrm {supp}}} Q$. We start by observing that a discrete chain rule yields

$$\begin{aligned}&\mathrm {div}^{\mathrm {d}}( {\widetilde{\Phi }}_n \circ \overline{\chi }_n^\perp ) = \nabla {\widetilde{\Phi }}_{1,n}(X_{n}) \cdot \partial ^{\mathrm {d}}_1 \overline{\chi }_n^\perp \nonumber \\&\quad + \nabla {\widetilde{\Phi }}_{2,n}(Y_n) \cdot \partial ^{\mathrm {d}}_2 \overline{\chi }_n^\perp \quad \text {in } \mathcal {PC}_{\lambda _n}(\mathbb {R}), \end{aligned}$$

(6.13)

where $(X_n)^{i,j}$ are vectors on the segment connecting $(\overline{\chi }_n^\perp )^{i,j}$ and $(\overline{\chi }_n^\perp )^{i+1,j}$ and the vectors $(Y_n)^{i,j}$ belong to the segment connecting $(\overline{\chi }_n^\perp )^{i,j}$ and $(\overline{\chi }_n^\perp )^{i,j+1}$. Suppose that $x \in \Omega ' \setminus \Omega _n^{\mathrm {supp}}$ and $x \in Q_{\lambda _n}(i,j)$. Let $(i',j') \in \mathbb {Z}^2$ be such that $|(i',j') - (i,j)| \leqq 1$ (possibly $(i',j') = (i,j)$) and $|\overline{\chi }_n^\perp | > M$ on $Q_{\lambda _n}(i',j')$ (thus $(\overline{\chi }_n^\perp )^{i',j'} \notin \mathrm {supp}({\widetilde{\Phi }}_n)$). Then we have that

$$\begin{aligned} \begin{aligned} |(X_n)^{i,j} - (\overline{\chi }_n^\perp )^{i',j'}|&\leqq |(X_n)^{i,j} - (\overline{\chi }_n^\perp )^{i,j}| + |(\overline{\chi }_n^\perp )^{i,j} - (\overline{\chi }_n^\perp )^{i',j'}| \\&\leqq C \lambda _n\big ( | \mathrm {D}^{\mathrm {d}}\overline{\chi }_n^{i,j}| + |\mathrm {D}^{\mathrm {d}}\overline{\chi }_n^{i-1,j} | + | \mathrm {D}^{\mathrm {d}}\overline{\chi }_n^{i,j-1}| \big ), \end{aligned} \end{aligned}$$

with a similar estimate being true for $Y_n$. Using that $\mathrm {D}{\widetilde{\Phi }}_n$ are equi-Lipschitz by (6.5) and that $\mathrm {D}{\widetilde{\Phi }}_n \big ( (\overline{\chi }_n^\perp )^{i',j'} \big ) = 0$, we get from (6.13) that

$$\begin{aligned} |\mathrm {div}^{\mathrm {d}}( {\widetilde{\Phi }}_n \circ \overline{\chi }_n^\perp ) (x)| \leqq C \lambda _n\big ( | \mathrm {D}^{\mathrm {d}}\overline{\chi }_n^{i,j}|^2 + |\mathrm {D}^{\mathrm {d}}\overline{\chi }_n^{i-1,j} |^2 + | \mathrm {D}^{\mathrm {d}}\overline{\chi }_n^{i,j-1}|^2 \big ). \end{aligned}$$

Fixing an open set $\Omega ''$ with $\Omega ' \subset \subset \Omega '' \subset \subset \Omega$ we conclude that for all n large enough

$$\begin{aligned} \bigg | \int _{\Omega ' \setminus \Omega _n^{\mathrm {supp}}} \zeta \mathrm {div}^{\mathrm {d}}( {\widetilde{\Phi }}_n \circ \overline{\chi }_n^\perp ) \, {\mathrm {d}{x}} \bigg | \leqq C \lambda _n\Vert \mathrm {D}^{\mathrm {d}}\overline{\chi }_n\Vert _{L^2(\Omega '')}^2 \leqq C \frac{\lambda _n}{\varepsilon _n} \rightarrow 0, \end{aligned}$$

where we have used (2.24) and (2.13). This implies (6.12).

Step 3. (Expanding the divergence using (6.6).) Let us observe first that after expanding $\mathrm {div}^{\mathrm {d}}({\widetilde{\Phi }}_n \circ \overline{\chi }_n^\perp )$ on $\Omega _n^{\mathrm {supp}}$ by (6.13), we can employ similar arguments as in Step 6 to replace the points $X_n$ and $Y_n$ in this formula by $\overline{\chi }_n^\perp$. Indeed, we have that $|X_n - \overline{\chi }_n^\perp |, |Y_n - \overline{\chi }_n^\perp | \leqq C \lambda _n|\mathrm {D}^{\mathrm {d}}\overline{\chi }_n|$ and using that $\mathrm {D}{\widetilde{\Phi }}_n$ are equi-Lipschitz and (2.24) we get that

$$\begin{aligned}&\int _{\Omega _n^{\mathrm {supp}}} \zeta \mathrm {div}^{\mathrm {d}}( {\widetilde{\Phi }}_n \circ \overline{\chi }_n^\perp ) \, {\mathrm {d}{x}} = \int _{\Omega _n^{\mathrm {supp}}} \zeta \big ( \nabla {\widetilde{\Phi }}_{1,n}(\overline{\chi }_n^\perp ) \cdot \partial ^{\mathrm {d}}_1 \overline{\chi }_n^\perp \nonumber \\&\quad + \nabla {\widetilde{\Phi }}_{2,n}(\overline{\chi }_n^\perp ) \cdot \partial ^{\mathrm {d}}_2 \overline{\chi }_n^\perp \big ) \, {\mathrm {d}{x}} + o_n(1). \end{aligned}$$

(6.14)

By (6.6) we get that

$$\begin{aligned} \begin{aligned}&\nabla {\widetilde{\Phi }}_{1,n}(\overline{\chi }_n^\perp ) \cdot \partial ^{\mathrm {d}}_1 \overline{\chi }_n^\perp + \nabla {\widetilde{\Phi }}_{2,n}(\overline{\chi }_n^\perp ) \cdot \partial ^{\mathrm {d}}_2 \overline{\chi }_n^\perp \\&\quad = \alpha _n(\overline{\chi }_n^\perp ) \big ( \partial ^{\mathrm {d}}_1 \overline{\chi }_{1,n}^\perp + \partial ^{\mathrm {d}}_2 \overline{\chi }_{2,n}^\perp \big ) \\&\qquad - q_n(\overline{\chi }_n) \big ( \nabla \Psi _{1,n}(\overline{\chi }_n^\perp ) \cdot \partial ^{\mathrm {d}}_1 \overline{\chi }_n^\perp + \nabla \Psi _{2,n}(\overline{\chi }_n^\perp ) \cdot \partial ^{\mathrm {d}}_2 \overline{\chi }_n^\perp \big ). \end{aligned} \end{aligned}$$

(6.15)

We now exploit the fact that $\mathrm {curl}^{\mathrm {d}}(\overline{\chi }_n)$ is approximately zero to obtain that

$$\begin{aligned} \int _{\Omega '} \big | \alpha _n(\overline{\chi }_n^\perp ) \big ( \partial ^{\mathrm {d}}_1 \overline{\chi }_{1,n}^\perp + \partial ^{\mathrm {d}}_2 \overline{\chi }_{2,n}^\perp \big ) \big | \, {\mathrm {d}{x}} = o_n(1) \rightarrow 0 \quad \text {as } n \rightarrow +\infty . \end{aligned}$$

(6.16)

Indeed, since $\big | \alpha _n(\overline{\chi }_n^\perp ) \big ( \partial ^{\mathrm {d}}_1 \overline{\chi }_{1,n}^\perp + \partial ^{\mathrm {d}}_2 \overline{\chi }_{2,n}^\perp \big ) \big | = |\alpha (\overline{\chi }_n^\perp )| |\mathrm {curl}^{\mathrm {d}}(\overline{\chi }_n)|$, this is a consequence of Lemma 2.4 and the fact that $\alpha _n$ are equibounded by (6.5). Combining (6.12), (6.14), (6.15), and (6.16), we have shown that

$$\begin{aligned} \begin{aligned}&\int _{\Omega '} \zeta \mathrm {div}^{\mathrm {d}}( {\widetilde{\Phi }}_n \circ \overline{\chi }_n^\perp ) \, {\mathrm {d}{x}} \\&\quad = - \int _{\Omega _n^{\mathrm {supp}}} \zeta q_n(\overline{\chi }_n) \Big ( \nabla \Psi _{1,n}(\overline{\chi }_n^\perp ) \cdot \partial ^{\mathrm {d}}_1 \overline{\chi }_n^\perp + \nabla \Psi _{2,n}(\overline{\chi }_n^\perp ) \cdot \partial ^{\mathrm {d}}_2 \overline{\chi }_n^\perp \Big ) \, {\mathrm {d}{x}} + o_n(1). \end{aligned}\nonumber \\ \end{aligned}$$

(6.17)

Next, we replace $\nabla \Psi _{1,n}(\overline{\chi }_n^\perp ) \cdot \partial ^{\mathrm {d}}_1 \overline{\chi }_n^\perp + \nabla \Psi _{2,n}(\overline{\chi }_n^\perp ) \cdot \partial ^{\mathrm {d}}_2 \overline{\chi }_n^\perp$ with $\mathrm {div}^{\mathrm {d}}(\Psi _n \circ {\widetilde{\chi }}_n^\perp )$ up to a small error, thus recovering the analogue of (6.3) in the discrete, cf. (6.21).¹² We start by using a discrete chain rule to get that

$$\begin{aligned} \mathrm {div}^{\mathrm {d}}(\Psi _n \circ {\widetilde{\chi }}_n^\perp ) = \nabla \Psi _{1,n}({\widetilde{X}}_n) \cdot \partial ^{\mathrm {d}}_1 {\widetilde{\chi }}_n^\perp + \nabla \Psi _{2,n}({\widetilde{Y}}_n) \cdot \partial ^{\mathrm {d}}_2 {\widetilde{\chi }}_n^\perp \end{aligned}$$

where $({\widetilde{X}}_n)^{i,j}$ belongs to the segment connecting $({\widetilde{\chi }}_n^\perp )^{i,j}$ and $({\widetilde{\chi }}_n^\perp )^{i+1,j}$, and $({\widetilde{Y}}_n)^{i,j}$ to the segment connecting $({\widetilde{\chi }}_n^\perp )^{i,j}$ and $({\widetilde{\chi }}_n^\perp )^{i,j+1}$. Using that $|\overline{\chi }_n^\perp | \leqq M$ on $\Omega _n^{\mathrm {supp}}$, that $q_n$ are locally equibounded, that $|{\widetilde{X}}_n - {\widetilde{\chi }}_n^\perp |, |{\widetilde{Y}}_n - {\widetilde{\chi }}_n^\perp | \leqq C \lambda _n|\mathrm {D}^{\mathrm {d}}{\widetilde{\chi }}_n|$, and that $\mathrm {D}\Psi _n$ are equi-Lipschitz by (6.5), we get that

$$\begin{aligned}&q_n(\overline{\chi }_n) \Big | \mathrm {div}^{\mathrm {d}}(\Psi _n \circ {\widetilde{\chi }}_n^\perp ) - \Big ( \nabla \Psi _{1,n}({\widetilde{\chi }}_n^\perp ) \cdot \partial ^{\mathrm {d}}_1 {\widetilde{\chi }}_n^\perp + \nabla \Psi _{2,n}({\widetilde{\chi }}_n^\perp ) \cdot \partial ^{\mathrm {d}}_2 {\widetilde{\chi }}_n^\perp \Big ) \Big | \\&\quad \leqq C \lambda _n|\mathrm {D}^{\mathrm {d}}{\widetilde{\chi }}_n|^2 \text { on } \Omega _n^{\mathrm {supp}}. \end{aligned}$$

By (2.26) and (2.13) we obtain that

$$\begin{aligned} \begin{aligned}&- \int _{\Omega _n^{\mathrm {supp}}} \zeta q_n(\overline{\chi }_n)\mathrm {div}^{\mathrm {d}}(\Psi _n \circ {\widetilde{\chi }}_n^\perp ) \, {\mathrm {d}{x}} \\&\quad = - \int _{\Omega _n^{\mathrm {supp}}} \zeta q_n(\overline{\chi }_n) \Big ( \nabla \Psi _{1,n}({\widetilde{\chi }}_n^\perp ) \cdot \partial ^{\mathrm {d}}_1 {\widetilde{\chi }}_n^\perp + \nabla \Psi _{2,n}({\widetilde{\chi }}_n^\perp ) \cdot \partial ^{\mathrm {d}}_2 {\widetilde{\chi }}_n^\perp \Big ) \, {\mathrm {d}{x}} + o_n(1). \end{aligned}\nonumber \\ \end{aligned}$$

(6.18)

Next, for $k=1,2$ we estimate

$$\begin{aligned}&\big | \nabla \Psi _{k,n}(\overline{\chi }_n^\perp ) \cdot \partial ^{\mathrm {d}}_k \overline{\chi }_n^\perp - \nabla \Psi _{k,n}({\widetilde{\chi }}_n^\perp ) \cdot \partial ^{\mathrm {d}}_k {\widetilde{\chi }}_n^\perp \big | \nonumber \\&\quad \leqq C | \overline{\chi }_n^\perp - {\widetilde{\chi }}_n^\perp | |\partial ^{\mathrm {d}}_k \overline{\chi }_n^\perp | + C|\partial ^{\mathrm {d}}_k \overline{\chi }_n^\perp - \partial ^{\mathrm {d}}_k {\widetilde{\chi }}_n^\perp |, \end{aligned}$$

(6.19)

where we have used again that $\mathrm {D}\Psi _n$ are equi-Lipschitz and equibounded. The term $| \overline{\chi }_n^\perp - {\widetilde{\chi }}_n^\perp |$ can be estimated as the difference $| \chi _n - \overline{\chi }_n|$ in Remark 2.5. Indeed, writing ${\widetilde{\chi }}_{h,n} = \frac{1}{\sqrt{\delta _n}} \sin (\sqrt{\delta _n} \overline{\chi }_{h,n})$ for $h=1,2$ and using that $|s - \sin (s)| \leqq C|s|^3$ we get $| \overline{\chi }_n^\perp - {\widetilde{\chi }}_n^\perp | \leqq C \delta _n|\overline{\chi }_n|^3$. Using a discrete chain rule in the above representation of ${\widetilde{\chi }}_{h,n}$ we also get that

$$\begin{aligned}&| \partial ^{\mathrm {d}}_k \overline{\chi }_{h,n} - \partial ^{\mathrm {d}}_k {\widetilde{\chi }}_{h,n} | = \big | 1 - \cos (\sqrt{\delta _n} X_{h,k,n}) \big | |\partial ^{\mathrm {d}}_k \overline{\chi }_{h,n}| \nonumber \\&\quad \leqq C \delta _n\big ( |\overline{\chi }_{h,n}|^2 + |\overline{\chi }_{h,n}^{{\varvec{\cdot }}+ e_k}|^2 \big ) |\partial ^{\mathrm {d}}_k \overline{\chi }_{h,n}|, \end{aligned}$$

(6.20)

for $k,h = 1,2$, where $X_{h,k,n}^{i,j}$ belongs to the segment connecting $\overline{\chi }_{h,n}^{(i,j)}$ and $\overline{\chi }_{h,n}^{(i,j) + e_k}$ and we have used that $| 1 - \cos (s)| \leqq C |s|^2$. Returning to (6.19) we now infer that

$$\begin{aligned} \begin{aligned}&\big | \nabla \Psi _{k,n}(\overline{\chi }_n^\perp ) \cdot \partial ^{\mathrm {d}}_k \overline{\chi }_n^\perp - \nabla \Psi _{k,n}({\widetilde{\chi }}_n^\perp ) \cdot \partial ^{\mathrm {d}}_k {\widetilde{\chi }}_n^\perp \big |\\&\quad \leqq C \delta _n|\partial ^{\mathrm {d}}_k \overline{\chi }_n| \big ( |\overline{\chi }_n|^3 + |\overline{\chi }_n|^2 + |\overline{\chi }_n^{{\varvec{\cdot }}+ e_k}|^2 \big ) \\&\quad \leqq CM^3 \delta _n|\mathrm {D}^{\mathrm {d}}\overline{\chi }_n| \qquad \text {on } \Omega _n^{\mathrm {supp}} \, , \end{aligned} \end{aligned}$$

cf. (6.11). Thus, by (6.17) and (6.18) we infer that

$$\begin{aligned} \begin{aligned}&\bigg | \int _{\Omega '} \zeta \mathrm {div}^{\mathrm {d}}( {\widetilde{\Phi }}_n \circ \overline{\chi }_n^\perp ) \, {\mathrm {d}{x}} + \int _{\Omega _n^{\mathrm {supp}}} \zeta q_n(\overline{\chi }_n)\mathrm {div}^{\mathrm {d}}(\Psi _n \circ {\widetilde{\chi }}_n^\perp ) \, {\mathrm {d}{x}} \bigg | \\&\quad \leqq CM^3 \delta _n\Vert q_n(\overline{\chi }_n)\Vert _{L^2(\Omega _n^{\mathrm {supp}})} \Vert \mathrm {D}^{\mathrm {d}}\overline{\chi }_n\Vert _{L^2(\Omega _n^{\mathrm {supp}})} + o_n(1)\\&\quad \leqq CM^3 \delta _n\frac{\sqrt{\varepsilon _n}}{\sqrt{\varepsilon _n}} + o_n(1) \rightarrow 0, \end{aligned} \end{aligned}$$

where we have used the fact that $q_n(\overline{\chi }_n)^2 = W(\chi _n)$, Proposition 2.6, and (2.24) in the proof thereof. In conclusion, we have proved that

$$\begin{aligned} \int _{\Omega '} \zeta \mathrm {div}^{\mathrm {d}}( {\widetilde{\Phi }}_n \circ \overline{\chi }_n^\perp \big ) \, {\mathrm {d}{x}} = - \int _{\Omega _n^{\mathrm {supp}}} \zeta q_n(\overline{\chi }_n) \mathrm {div}^{\mathrm {d}}(\Psi _n \circ {\widetilde{\chi }}_n^\perp ) \, {\mathrm {d}{x}} + o_n(1). \end{aligned}$$

(6.21)

Step 4. (Young’s inequality.) Applying Young’s inequality to (6.21) we get that

$$\begin{aligned}&\int _{\Omega '} \zeta \mathrm {div}^{\mathrm {d}}( {\widetilde{\Phi }}_n \circ \overline{\chi }_n^\perp ) \, {\mathrm {d}{x}} \leqq \frac{1}{2}\int _{\Omega _n^{\mathrm {supp}}} \frac{1}{\varepsilon _n} q_n(\overline{\chi }_n)^2 \, {\mathrm {d}{x}} \nonumber \\&\quad + \frac{1}{2} \int _{\Omega '} \varepsilon _n|\zeta |^2 |\mathrm {div}^{\mathrm {d}}(\Psi _n \circ {\widetilde{\chi }}_n^\perp )|^2 \, {\mathrm {d}{x}} + o_n(1). \end{aligned}$$

(6.22)

Step 5. (Recovering the potential term.) We prove that

$$\begin{aligned} \frac{1}{\varepsilon _n} \int _{\Omega _n^{\mathrm {supp}}} q_n(\overline{\chi }_n)^2 \, {\mathrm {d}{x}} \leqq \frac{1}{\varepsilon _n} \int _{\Omega '} W^\mathrm {d}(\chi _n) \, {\mathrm {d}{x}} + o_n(1). \end{aligned}$$

(6.23)

We proceed similarly as in Step 2.6 of the proof of Proposition 2.6: By (2.36) we have that

$$\begin{aligned} \big | \sqrt{W^\mathrm {d}}(\chi _n^{i,j}) - \sqrt{W}(\chi _n^{i,j}) \big | \leqq C M \lambda _n\big ( |\mathrm {D}^{\mathrm {d}}\chi _n^{i-1,j}| + |\mathrm {D}^{\mathrm {d}}\chi _n^{i,j-1}| \big ) \quad \text {on } \Omega _n^{\mathrm {supp}} \end{aligned}$$

according to (6.11), where we have used that $|\chi _n| \leqq |\overline{\chi }_n|$. (This is seen by using in (2.8) the fact that $|\sin (x)| \leqq |x|$.) Let $\Omega ''$ again be an open set with $\Omega ' \subset \subset \Omega '' \subset \subset \Omega$. By the bound (2.25) and by (2.13) we obtain for n large enough that

$$\begin{aligned} \frac{1}{\sqrt{\varepsilon _n}} \big \Vert \sqrt{W^\mathrm {d}}(\chi _n) - \sqrt{W}(\chi _n) \big \Vert _{L^2(\Omega _n^{\mathrm {supp}})} \leqq CM \frac{\lambda _n}{\sqrt{\varepsilon _n}} \Vert \mathrm {D}^{\mathrm {d}}\chi _n \Vert _{L^2(\Omega '')} \leqq CM \frac{\lambda _n}{\varepsilon _n} \rightarrow 0. \end{aligned}$$

Since $W^\mathrm {d}- W = \big ( 2 \sqrt{W^\mathrm {d}} - (\sqrt{W^\mathrm {d}} - \sqrt{W}) \big ) \big ( \sqrt{W^\mathrm {d}} - \sqrt{W} \big )$ and $q_n(\overline{\chi }_n)^2 = W(\chi _n)$ we get that

$$\begin{aligned} \begin{aligned}&\frac{1}{\varepsilon _n} \int _{\Omega _n^{\mathrm {supp}}} |W^\mathrm {d}(\chi _n) - q_n(\overline{\chi }_n)^2| \, {\mathrm {d}{x}} \\&\quad \leqq \Big ( \tfrac{2}{\sqrt{\varepsilon _n}} \big \Vert \sqrt{W^\mathrm {d}}(\chi _n) \big \Vert _{L^2(\Omega _n^{\mathrm {supp}})} + o_n(1) \Big ) \cdot o_n(1) \rightarrow 0, \end{aligned} \end{aligned}$$

where we have used that $\Vert W^\mathrm {d}(\chi _n)\Vert _{L^2(\Omega )} \leqq C \sqrt{\varepsilon _n}$. Since $\Omega _n^{\mathrm {supp}} \subset \Omega '$ and $W^\mathrm {d}\geqq 0$, we conclude the proof of (6.23).

Step 6. (From discrete divergence to full discrete derivative matrix.) In the next steps we recover the derivative term $|A^\mathrm {d}(\chi _n)|^2$. We start by claiming that

$$\begin{aligned}&\varepsilon _n\int _{\Omega '} |\zeta |^2 |\mathrm {div}^{\mathrm {d}}(\Psi _n \circ {\widetilde{\chi }}_n^\perp )|^2 \, {\mathrm {d}{x}} \nonumber \\&\quad \leqq \varepsilon _n\int _{\Omega '} | \zeta |^2 |\mathrm {D}^{\mathrm {d}}(\Psi _n \circ {\widetilde{\chi }}_n^\perp )|^2 \, {\mathrm {d}{x}} + o_n(1), \end{aligned}$$

(6.24)

see (6.4) for the analogous inequality in the continuum. Let us use the short-hand notation $V_n := \Psi _n \circ {\widetilde{\chi }}_n^\perp$. We prove the claim first with a perturbed version of $\mathrm {D}^{\mathrm {d}}(\Psi _n \circ {\widetilde{\chi }}_n^\perp )$, where we add certain shifts in the lattice point. Specifically, let us observe that

$$\begin{aligned} \begin{aligned}&|\partial ^{\mathrm {d}}_1 V_{1,n}|^2 + |\partial ^{\mathrm {d}}_1 V_{2,n}^{{\varvec{\cdot }}+ e_2}|^2 + |\partial ^{\mathrm {d}}_2 V_{1,n}^{{\varvec{\cdot }}+ e_1}|^2 + |\partial ^{\mathrm {d}}_2 V_{2,n}|^2 \\&\quad = |\mathrm {div}^{\mathrm {d}}(V_n)|^2 + |\partial ^{\mathrm {d}}_1 V_{2,n}^{{\varvec{\cdot }}+ e_2} - \partial ^{\mathrm {d}}_2 V_{1,n}^{{\varvec{\cdot }}+ e_1}|^2 \\&\qquad - 2 \partial ^{\mathrm {d}}_1 (V_{1,n} \partial ^{\mathrm {d}}_2 V_{2,n}) + 2 \partial ^{\mathrm {d}}_2 (V_{1,n}^{{\varvec{\cdot }}+ e_1} \partial ^{\mathrm {d}}_1 V_{2,n}), \end{aligned} \end{aligned}$$

(6.25)

because $- \partial ^{\mathrm {d}}_1 (V_{1,n} \partial ^{\mathrm {d}}_2 V_{2,n}) + \partial ^{\mathrm {d}}_2 (V_{1,n}^{{\varvec{\cdot }}+ e_1} \partial ^{\mathrm {d}}_1 V_{2,n}) = - \partial ^{\mathrm {d}}_1 V_{1,n} \partial ^{\mathrm {d}}_2 V_{2,n} + \partial ^{\mathrm {d}}_2 V_{1,n}^{{\varvec{\cdot }}+ e_1} \partial ^{\mathrm {d}}_1 V_{2,n}^{{\varvec{\cdot }}+ e_2}$ by the discrete product rule. Since $\zeta$ is compactly supported in $\Omega$, a discrete integration by parts allows us to conclude that

$$\begin{aligned} \begin{aligned}&\varepsilon _n\! \int _{\Omega '} |\zeta |^2 |\mathrm {div}^{\mathrm {d}}(V_n)|^2 \, {\mathrm {d}{x}} \\&\quad \leqq \varepsilon _n\! \int _{\Omega '} |\zeta |^2 \big ( |\partial ^{\mathrm {d}}_1 V_{1,n}|^2 + |\partial ^{\mathrm {d}}_1 V_{2,n}^{{\varvec{\cdot }}+ e_2}|^2 + |\partial ^{\mathrm {d}}_2 V_{1,n}^{{\varvec{\cdot }}+ e_1}|^2 + |\partial ^{\mathrm {d}}_2 V_{2,n}|^2 \big ) \, {\mathrm {d}{x}} \\&\qquad - 2 \varepsilon _n\! \int _{\Omega '} \partial ^{\mathrm {d}}_{1}(|\zeta |^2) \, (V_{1,n} \partial ^{\mathrm {d}}_2 V_{2,n})^{{\varvec{\cdot }}+ e_1} \! - \partial ^{\mathrm {d}}_{2}(|\zeta |^2) \, (V_{1,n}^{{\varvec{\cdot }}+ e_1} \partial ^{\mathrm {d}}_1 V_{2,n})^{{\varvec{\cdot }}+ e_2} \, {\mathrm {d}{x}} \end{aligned} \end{aligned}$$

(6.26)

for all n large enough. Notice that, even although $|\zeta |^2$ is not a discrete function, it is still possible to use a discrete integration by parts when we extend the notion of discrete derivatives to non-discrete functions by making use of difference quotients. Specifically, for any function f on $\mathbb {R}^2$ we set $\partial ^{\mathrm {d}}_{k} f(x) := \frac{1}{\lambda _n} (f(x + \lambda _ne_k) - f(x))$ for $k = 1,2$, where it will always be clear from the context which lattice spacing $\lambda _n$ is to be considered. Since $\Psi _n$ are equibounded, we have that $|V_n| \leqq C$ with C independent of n. Since moreover $|\partial ^{\mathrm {d}}_{k}(|\zeta |^2)| \leqq \Vert \nabla (|\zeta |^2) \Vert _{L^\infty }$, we can estimate the modulus of the last integral above by $C \varepsilon _n\Vert \mathrm {D}^{\mathrm {d}}V_n \Vert _{L^1(\Omega ')}$ for n large enough. Since $\Psi _n$ are equi-Lipschitz, this can be further estimated by $C \varepsilon _n\Vert \mathrm {D}^{\mathrm {d}}{\widetilde{\chi }}_n \Vert _{L^2(\Omega ')}$ which goes to zero, since $\Vert \mathrm {D}^{\mathrm {d}}{\widetilde{\chi }}_n \Vert _{L^2(\Omega ')} \leqq \frac{C}{\sqrt{\varepsilon _n}}$by (2.26). Using a shift of variables we now obtain that

$$\begin{aligned} \begin{aligned} \varepsilon _n\int _{\Omega '} |\zeta |^2 |\mathrm {div}^{\mathrm {d}}(V_n)|^2 \, {\mathrm {d}{x}}&\leqq \varepsilon _n\int _{\Omega '} |\zeta |^2 |\mathrm {D}^{\mathrm {d}}V_n|^2 \, {\mathrm {d}{x}} + o_n(1) \\&\quad + \varepsilon _n\int _{\Omega '} \big ( |\zeta (x - \lambda _ne_2)|^2 - |\zeta (x)|^2 \big ) |\partial ^{\mathrm {d}}_1 V_{2,n}|^2 \\&\quad + \big ( |\zeta (x - \lambda _ne_1)|^2 - |\zeta (x)|^2 \big ) |\partial ^{\mathrm {d}}_2 V_{1,n}|^2 \, {\mathrm {d}{x}}. \end{aligned} \end{aligned}$$

Using that $|\zeta |^2$ is Lipschitz, that $\Psi _n$ are equi-Lipschitz, and (2.24), we can estimate the last integral above by $C\lambda _n= o_n(1)$. Thus we obtain (6.24).

Step 7. (Identifying “bad” cells.) Next, we want to use the inequality

$$\begin{aligned} |\mathrm {D}^{\mathrm {d}}(\Psi _n \circ {\widetilde{\chi }}_n^\perp )|^2 \leqq \mathrm {Lip}(\Psi _n)^2 |\mathrm {D}^{\mathrm {d}}{\widetilde{\chi }}_n|^2, \end{aligned}$$

and afterwards the fact that $\mathrm {curl}^{\mathrm {d}}({\widetilde{\chi }}_n)$ is approximately zero to later recover the term $|A^\mathrm {d}(\chi _n)|^2$ (a discrete divergence of ${\widetilde{\chi }}_n$ with shifts in the lattice points) from $|\mathrm {D}^{\mathrm {d}}{\widetilde{\chi }}_n|^2$. However, before dismissing the approximations $\Psi _n$, we need to exploit their uniform boundedness in cells where $\mathrm {curl}^{\mathrm {d}}({\widetilde{\chi }}_n)$ is large. (This step can be avoided under the additional scaling assumption $\frac{\delta _n^{3/2}}{\lambda _n} \rightarrow 0$, cf. Footnote 13.)

For a small parameter $t > 0$ (we take $t < \frac{\pi }{2}$, see below), we introduce the collection of bad cells

$$\begin{aligned} \begin{aligned} \mathcal {Q}^{\mathrm {bad}}_{n,t} := \big \{ Q_{\lambda _n}(i,j)&\ : \ (i,j) \in \mathbb {Z}^2, \ Q_{\lambda _n}(i,j) \cap \Omega ' \ne \emptyset , \\&\qquad |(\overline{\chi }_n)^{i',j'}| > \tfrac{t}{\sqrt{\delta _n}} \text { for some vertex } \lambda _n(i',j') \text { of } Q_{\lambda _n}(i,j) \big \} \end{aligned} \end{aligned}$$

and the set

$$\begin{aligned} \Omega ^{\mathrm {bad}}_{n,t}:= \bigcup _{Q \in \mathcal {Q}^{\mathrm {bad}}_{n,t}} Q. \end{aligned}$$

We note that we have an $L^2$ control on $\mathrm {curl}^{\mathrm {d}}({\widetilde{\chi }}_n)$ on the remaining set $\Omega ' \setminus \Omega ^{\mathrm {bad}}_{n,t}$. To see this, let us recall from the proof of Lemma 2.4 that under the assumption (2.20) we have that $\mathrm {curl}^{\mathrm {d}}(\overline{\chi }_n)^{i,j} = 0$. As a consequence, recalling the definition (2.19) we have that $\mathrm {curl}^{\mathrm {d}}(\overline{\chi }_n) \equiv 0$ on $\Omega ' \setminus \Omega ^{\mathrm {bad}}_{n,t}$ if $t < \frac{\pi }{2}$. Then, as in (6.20) we get that

$$\begin{aligned}&|\mathrm {curl}^{\mathrm {d}}({\widetilde{\chi }}_n)| = |\mathrm {curl}^{\mathrm {d}}({\widetilde{\chi }}_n) - \mathrm {curl}^{\mathrm {d}}(\overline{\chi }_n)| \\&\quad \leqq C \delta _n\big ( |\overline{\chi }_n|^2 + |\overline{\chi }_n^{{\varvec{\cdot }}+ e_1}|^2 + |\overline{\chi }_n^{{\varvec{\cdot }}+ e_2}|^2 ) |\mathrm {D}^{\mathrm {d}}\overline{\chi }_n| \leqq Ct^2 |\mathrm {D}^{\mathrm {d}}\overline{\chi }_n| \end{aligned}$$

on $\Omega ' \setminus \Omega ^{\mathrm {bad}}_{n,t}$. Using the estimate (2.24) from the proof of Proposition 2.6 we conclude that

$$\begin{aligned} \varepsilon _n\int _{\Omega ' \setminus \Omega ^{\mathrm {bad}}_{n,t}} |\mathrm {curl}^{\mathrm {d}}({\widetilde{\chi }}_n)|^2 \, {\mathrm {d}{x}} \leqq C t^4. \end{aligned}$$

(6.27)

We also note that we have an estimate on the number of bad cells: In view of Lemma 2.3, the number of vertices $\lambda _n(i',j')$ with $|(\overline{\chi }_n)^{i',j'}| > \tfrac{t}{\sqrt{\delta _n}}$ and such that $Q_{\lambda _n}(i',j') \subset \Omega$ is at most $C(t) \frac{\delta _n^{3/2}}{\lambda _n}$. Since $\Omega ' \subset \subset \Omega$, for large enough n, $\# \mathcal {Q}^{\mathrm {bad}}_{n,t}$ is at most four times larger. Thus, $\# \mathcal {Q}^{\mathrm {bad}}_{n,t} \leqq C(t) \frac{\delta _n^{3/2}}{\lambda _n}$.¹³

Step 8. (Removing a neighborhood of the “bad” cells.) We introduce a neighborhood of the “bad” cells

$$\begin{aligned} N_{n,t} := \Omega ^{\mathrm {bad}}_{n,t} + B_{r_n}, \end{aligned}$$

where $B_{r_n}$ is the ball centered at 0 with radius $r_n$. If $r_n \ll \varepsilon _n$ (e.g., $r_n := \delta _n^{1/4} \varepsilon _n$) we claim that

$$\begin{aligned}&\varepsilon _n\int _{\Omega '} |\zeta |^2 |\mathrm {D}^{\mathrm {d}}(\Psi _n \circ {\widetilde{\chi }}_n^\perp )|^2 \, {\mathrm {d}{x}}\nonumber \\&\quad \leqq \mathrm {Lip}(\Psi _n)^2 \varepsilon _n\int _{\Omega ' \setminus N_{n,t}} |\zeta |^2 |\mathrm {D}^{\mathrm {d}}{\widetilde{\chi }}_n|^2 \, {\mathrm {d}{x}} + o_n(1) . \end{aligned}$$

(6.28)

Indeed, we have that $\varepsilon _n\int _{N_{n,t}} |\zeta |^2 |\mathrm {D}^{\mathrm {d}}(\Psi _n \circ {\widetilde{\chi }}_n^\perp )|^2 \, {\mathrm {d}{x}} \rightarrow 0$ and $|\mathrm {D}^{\mathrm {d}}(\Psi _n \circ {\widetilde{\chi }}_n^\perp )| \leqq \mathrm {Lip}(\Psi _n) |\mathrm {D}^{\mathrm {d}}{\widetilde{\chi }}_n|$. To prove the former, let us note that for every $Q \in \mathcal {Q}^{\mathrm {bad}}_{n,t}$, $Q + B_{r_n}$ is contained in a ball of radius $r_n + \frac{\lambda _n}{\sqrt{2}}$. Thus, $\mathcal {L}^2(N_{n,t}) \leqq \# \mathcal {Q}^{\mathrm {bad}}_{n,t} \pi \big (r_n + \frac{\lambda _n}{\sqrt{2}} \big )^2 \leqq C(t) \frac{\delta _n^{3/2}}{\lambda _n}(r_n^2 + \lambda _n^2)$, where we have used the estimate on $\# \mathcal {Q}^{\mathrm {bad}}_{n,t}$ derived in Step 6. Moreover, we have that $|\mathrm {D}^{\mathrm {d}}(\Psi _n \circ {\widetilde{\chi }}_n^\perp )| \leqq \frac{C}{\lambda _n}$ since $\Psi _n$ are bounded in $L^\infty$. In conclusion, using (2.13),

$$\begin{aligned}&\varepsilon _n\int _{N_{n,t}} |\zeta |^2 |\mathrm {D}^{\mathrm {d}}(\Psi _n \circ {\widetilde{\chi }}_n^\perp )|^2 \, {\mathrm {d}{x}} \\&\quad \leqq C(t) \varepsilon _n\frac{\delta _n^{3/2}}{\lambda _n} (r_n^2 + \lambda _n^2) \frac{1}{\lambda _n^2} = C(t) \frac{r_n^2 + \lambda _n^2}{\varepsilon _n^2} \rightarrow 0. \end{aligned}$$

Step 9. (From full discrete derivative matrix to $A^\mathrm {d}$, outside “bad” cells.) To ease the integration by parts, we introduce cut-off functions $\rho _{n,t} \in C^\infty (\mathbb {R}^2;[0,1])$ such that $\rho _{n,t} = 0$ in $\Omega ^{\mathrm {bad}}_{n,t}$, $\rho _{n,t} = 1$ in $\Omega ' \setminus \overline{N}_{n,t}$, and $|\nabla \rho _{n,t}| \leqq \tfrac{C}{r_n}$. We set $\eta _{n,t} := \rho _{n,t} |\zeta |^2$. By (6.28) we have that

$$\begin{aligned} \varepsilon _n\int _{\Omega '} |\zeta |^2 |\mathrm {D}^{\mathrm {d}}(\Psi _n \circ {\widetilde{\chi }}_n^\perp )|^2 \, {\mathrm {d}{x}} \leqq \mathrm {Lip}(\Psi _n)^2 \varepsilon _n\int _{\Omega '} \eta _{n,t} |\mathrm {D}^{\mathrm {d}}{\widetilde{\chi }}_n|^2 \, {\mathrm {d}{x}} + o_n(1) \, . \end{aligned}$$

(6.29)

Let us observe that, similarly to (6.25),

$$\begin{aligned} \begin{aligned}&|\partial ^{\mathrm {d}}_1 {\widetilde{\chi }}^{{\varvec{\cdot }}- e_1}_{1,n}|^2 + |\partial ^{\mathrm {d}}_1 {\widetilde{\chi }}_{2,n}|^2 + |\partial ^{\mathrm {d}}_2 {\widetilde{\chi }}_{1,n}|^2 + |\partial ^{\mathrm {d}}_2 {\widetilde{\chi }}^{{\varvec{\cdot }}- e_2}_{2,n}|^2 \\&\quad = \big |\partial ^{\mathrm {d}}_1 {\widetilde{\chi }}^{{\varvec{\cdot }}- e_1}_{1,n} + \partial ^{\mathrm {d}}_2 {\widetilde{\chi }}^{{\varvec{\cdot }}- e_2}_{2,n} \big |^2 + |\mathrm {curl}^{\mathrm {d}}({\widetilde{\chi }}_n)|^2 - 2 \partial ^{\mathrm {d}}_1 \big ({\widetilde{\chi }}^{{\varvec{\cdot }}- e_1}_{1,n} \partial ^{\mathrm {d}}_2 {\widetilde{\chi }}^{{\varvec{\cdot }}- e_2}_{2,n} \big )\\&\qquad + 2 \partial ^{\mathrm {d}}_2 \big ({\widetilde{\chi }}_{1,n} \partial ^{\mathrm {d}}_2 {\widetilde{\chi }}^{{\varvec{\cdot }}- e_2}_{2,n} \big ), \end{aligned} \end{aligned}$$

and note that $\big |\partial ^{\mathrm {d}}_1 {\widetilde{\chi }}^{{\varvec{\cdot }}- e_1}_{1,n} + \partial ^{\mathrm {d}}_2 {\widetilde{\chi }}^{{\varvec{\cdot }}- e_2}_{2,n} \big |^2 = |A^\mathrm {d}(\chi _n)|^2$. Thus, by shifting variables and using a discrete integration by parts, we get that, for n large enough,

$$\begin{aligned} \begin{aligned}&\varepsilon _n\int _{\Omega '} \eta _{n,t} |\mathrm {D}^{\mathrm {d}}{\widetilde{\chi }}_n|^2 \, {\mathrm {d}{x}} \\&\quad = \varepsilon _n\int _{\Omega '} \eta _{n,t} \big ( |\partial ^{\mathrm {d}}_1 {\widetilde{\chi }}^{{\varvec{\cdot }}- e_1}_{1,n}|^2 + |\partial ^{\mathrm {d}}_1 {\widetilde{\chi }}_{2,n}|^2 + |\partial ^{\mathrm {d}}_2 {\widetilde{\chi }}_{1,n}|^2 + |\partial ^{\mathrm {d}}_2 {\widetilde{\chi }}^{{\varvec{\cdot }}- e_2}_{2,n}|^2 \big ) \, {\mathrm {d}{x}} \\&\qquad + \varepsilon _n\int _{\Omega '} |\partial ^{\mathrm {d}}_1 {\widetilde{\chi }}_{1,n}|^2 \big ( \eta _{n,t}(x) - \eta _{n,t}(x + \lambda _ne_1) \big ) \\&\qquad + |\partial ^{\mathrm {d}}_2 {\widetilde{\chi }}_{2,n}|^2 \big ( \eta _{n,t}(x) - \eta _{n,t}(x + \lambda _ne_2) \big ) \, {\mathrm {d}{x}} \\&\quad = \varepsilon _n\int _{\Omega '} \eta _{n,t} |A^\mathrm {d}(\chi _n)|^2 \, {\mathrm {d}{x}} + \varepsilon _n\int _{\Omega '} \eta _{n,t} |\mathrm {curl}^{\mathrm {d}}({\widetilde{\chi }}_n)|^2 \, {\mathrm {d}{x}} \\&\qquad + 2 \varepsilon _n\int _{\Omega '} \partial ^{\mathrm {d}}_{1}\eta _{n,t} \big ({\widetilde{\chi }}^{{\varvec{\cdot }}- e_1}_{1,n} \partial ^{\mathrm {d}}_2 {\widetilde{\chi }}^{{\varvec{\cdot }}- e_2}_{2,n} \big )^{{\varvec{\cdot }}+ e_1} - \partial ^{\mathrm {d}}_{2}\eta _{n,t} \big ({\widetilde{\chi }}_{1,n} \partial ^{\mathrm {d}}_2 {\widetilde{\chi }}^{{\varvec{\cdot }}- e_2}_{2,n} \big )^{{\varvec{\cdot }}+ e_2} \, {\mathrm {d}{x}} \\&\qquad - \varepsilon _n\int _{\Omega '} |\partial ^{\mathrm {d}}_1 {\widetilde{\chi }}_{1,n}|^2 \lambda _n\partial ^{\mathrm {d}}_{1}\eta _{n,t} + |\partial ^{\mathrm {d}}_2 {\widetilde{\chi }}_{2,n}|^2 \lambda _n\partial ^{\mathrm {d}}_{2}\eta _{n,t} \, {\mathrm {d}{x}}, \end{aligned} \end{aligned}$$

(6.30)

where as in (6.26) we let $\partial ^{\mathrm {d}}_{1} \eta _{n,t}$, $\partial ^{\mathrm {d}}_{2} \eta _{n,t}$ denote difference quotients of the function $\eta _{n,t}$. By (6.27) we have that

$$\begin{aligned} \varepsilon _n\int _{\Omega '} \eta _{n,t} |\mathrm {curl}^{\mathrm {d}}({\widetilde{\chi }}_n)|^2 \, {\mathrm {d}{x}} \leqq C t^4. \end{aligned}$$

(6.31)

Moreover, since $\varepsilon _n\int _{\Omega '} |\mathrm {D}^{\mathrm {d}}{\widetilde{\chi }}_n|^2 \, {\mathrm {d}{x}} \leqq C$ (by (2.26)) and $|\partial ^{\mathrm {d}}_{k} \eta _{n,t}| \leqq \Vert \nabla \eta _{n,t} \Vert _{L^\infty } \leqq \frac{C}{r_n}$, we obtain that

$$\begin{aligned} \bigg | \varepsilon _n\int _{\Omega '} |\partial ^{\mathrm {d}}_1 {\widetilde{\chi }}_{1,n}|^2 \lambda _n\partial ^{\mathrm {d}}_{1}\eta _{n,t} + |\partial ^{\mathrm {d}}_2 {\widetilde{\chi }}_{2,n}|^2 \lambda _n\partial ^{\mathrm {d}}_{2}\eta _{n,t} \, {\mathrm {d}{x}} \bigg | \leqq C \frac{\lambda _n}{r_n} \rightarrow 0, \end{aligned}$$

(6.32)

provided we choose $r_n$ such that $\lambda _n\ll r_n$, e.g., $r_n = \delta _n^{1/4} \varepsilon _n$ as proposed in Step 6. Finally, we show that

$$\begin{aligned}&2 \varepsilon _n\int _{\Omega '} \partial ^{\mathrm {d}}_{1}\eta _{n,t} \big ({\widetilde{\chi }}^{{\varvec{\cdot }}- e_1}_{1,n} \partial ^{\mathrm {d}}_2 {\widetilde{\chi }}^{{\varvec{\cdot }}- e_2}_{2,n} \big )^{{\varvec{\cdot }}+ e_1} \nonumber \\&\quad - \partial ^{\mathrm {d}}_{2}\eta _{n,t} \big ({\widetilde{\chi }}_{1,n} \partial ^{\mathrm {d}}_2 {\widetilde{\chi }}^{{\varvec{\cdot }}- e_2}_{2,n} \big )^{{\varvec{\cdot }}+ e_2} \, {\mathrm {d}{x}} = C(t) o_n(1). \end{aligned}$$

(6.33)

To this end, let us use that $|\partial ^{\mathrm {d}}_{k} \eta _{n,t}(x)| \leqq \Vert \nabla \eta _{n,t} \Vert _{L^\infty (B_{\lambda _n}(x))}$ and that $\nabla \eta _{n,t} = \rho _{n,t} \nabla (|\zeta |^2) + |\zeta |^2 \nabla \rho _{n,t}$ is bounded by C on $\Omega ' \setminus N_{n,t}$ and by $\frac{C}{r_n}$ on $N_{n,t}$. As a consequence, for n large enough,

$$\begin{aligned} \begin{aligned}&\bigg | 2 \varepsilon _n\int _{\Omega '} \partial ^{\mathrm {d}}_{1}\eta _{n,t} \big ({\widetilde{\chi }}^{{\varvec{\cdot }}- e_1}_{1,n} \partial ^{\mathrm {d}}_2 {\widetilde{\chi }}^{{\varvec{\cdot }}- e_2}_{2,n} \big )^{{\varvec{\cdot }}+ e_1} - \partial ^{\mathrm {d}}_{2}\eta _{n,t} \big ({\widetilde{\chi }}_{1,n} \partial ^{\mathrm {d}}_2 {\widetilde{\chi }}^{{\varvec{\cdot }}- e_2}_{2,n} \big )^{{\varvec{\cdot }}+ e_2} \, {\mathrm {d}{x}} \bigg | \\&\qquad \leqq C \frac{\varepsilon _n}{r_n} \int _{N_{n,t} + B_{\lambda _n}} |{\widetilde{\chi }}_{1,n}| |\partial ^{\mathrm {d}}_2 {\widetilde{\chi }}_{2,n}^{{\varvec{\cdot }}+ e_1 - e_2}|+ |{\widetilde{\chi }}_{1,n}^{{\varvec{\cdot }}+ e_2}| |\partial ^{\mathrm {d}}_1 {\widetilde{\chi }}_{2,n}| \, {\mathrm {d}{x}} \\&\quad \qquad + C \varepsilon _n\Vert {\widetilde{\chi }}_{1,n} \Vert _{L^2(\Omega ')} \Vert \mathrm {D}^{\mathrm {d}}{\widetilde{\chi }}_{2,n} \Vert _{L^2(\Omega ')}. \end{aligned} \end{aligned}$$

To further estimate this expression, let us observe that the function $s \mapsto |s| - \sqrt{(|s|^2 - 1)^+}$ belongs to $C_0(\mathbb {R}; \mathbb {R})$ and, as a consequence, is bounded. Since $|{\widetilde{\chi }}_{1,n}| \leqq |\chi _{1,n}|$ by their definition (2.8), we get that $|{\widetilde{\chi }}_{1,n}| \leqq \sqrt{(|\chi _{1,n}|^2 - 1)^+} + C$. Using Hölder’s inequality, this allows us to infer that

$$\begin{aligned} \begin{aligned}&\bigg | 2 \varepsilon _n\int _{\Omega '} \partial ^{\mathrm {d}}_{1}\eta _{n,t} \big ({\widetilde{\chi }}^{{\varvec{\cdot }}- e_1}_{1,n} \partial ^{\mathrm {d}}_2 {\widetilde{\chi }}^{{\varvec{\cdot }}- e_2}_{2,n} \big )^{{\varvec{\cdot }}+ e_1} - \partial ^{\mathrm {d}}_{2}\eta _{n,t} \big ({\widetilde{\chi }}_{1,n} \partial ^{\mathrm {d}}_2 {\widetilde{\chi }}^{{\varvec{\cdot }}- e_2}_{2,n} \big )^{{\varvec{\cdot }}+ e_2} \, {\mathrm {d}{x}} \bigg | \\&\qquad \leqq C \varepsilon _n\Vert \mathrm {D}^{\mathrm {d}}{\widetilde{\chi }}_{2,n}\Vert _{L^2(\Omega ')} \Big (\frac{1}{r_n} \big ( \mathcal {L}^2( N_{n,t} + B_{3 \lambda _n}) \big )^{1/4} \big \Vert \sqrt{(|\chi _{1,n}|^2-1)^+} \big \Vert _{L^4(\Omega ')} \\&\qquad \quad + \frac{1}{r_n} \big ( \mathcal {L}^2( N_{n,t} + B_{3 \lambda _n}) \big )^{1/2} + \Vert {\widetilde{\chi }}_{1,n} \Vert _{L^2(\Omega ')} \Big ) \end{aligned} \end{aligned}$$

for n large enough. We recall that $\Vert \mathrm {D}^{\mathrm {d}}{\widetilde{\chi }}_{2,n}\Vert _{L^2(\Omega ')} \leqq \frac{C}{\sqrt{\varepsilon _n}}$ by (2.26). We observe moreover that $\big \Vert \sqrt{(|\chi _{1,n}|^2-1)^+} \big \Vert _{L^4(\Omega ')} \leqq \Vert W(\chi _n) \Vert _{L^1(\Omega ')}^{1/4} \leqq C \varepsilon _n^{1/4}$ by Proposition 2.6. Furthermore, we have that $\Vert {\widetilde{\chi }}_{1,n} \Vert _{L^2(\Omega ')}\leqq C$ by Remark 2.2. Finally, since $\lambda _n\ll r_n$ (according to our choice of $r_n$) and in view of the bound on $\# \mathcal {Q}^{\mathrm {bad}}_{n,t}$ from Step 6 we have that $\mathcal {L}^2( N_{n,t} + B_{3 \lambda _n}) \leqq \# \mathcal {Q}^{\mathrm {bad}}_{n,t} C r_n^2 \leqq C(t) \frac{\delta _n^{3/2} r_n^2}{\lambda _n}$. Therefore, using (2.13) we obtain that

$$\begin{aligned} \begin{aligned}&\bigg | 2 \varepsilon _n\int _{\Omega '} \partial ^{\mathrm {d}}_{1}\eta _{n,t} \big ({\widetilde{\chi }}^{{\varvec{\cdot }}- e_1}_{1,n} \partial ^{\mathrm {d}}_2 {\widetilde{\chi }}^{{\varvec{\cdot }}- e_2}_{2,n} \big )^{{\varvec{\cdot }}+ e_1} - \partial ^{\mathrm {d}}_{2}\eta _{n,t} \big ({\widetilde{\chi }}_{1,n} \partial ^{\mathrm {d}}_2 {\widetilde{\chi }}^{{\varvec{\cdot }}- e_2}_{2,n} \big )^{{\varvec{\cdot }}+ e_2} \, {\mathrm {d}{x}} \bigg | \\&\qquad \leqq C(t) \sqrt{\varepsilon _n} \Big ( \tfrac{\delta _n^{3/8} \varepsilon _n^{1/4}}{r_n^{1/2} \lambda _n^{1/4}} + \tfrac{\delta _n^{3/4}}{\lambda _n^{1/2}} + 1 \Big ) = C(t) \Big ( \tfrac{\lambda _n^{1/2}}{r_n^{1/2}} + \delta _n^{1/2} + \varepsilon _n^{1/2} \Big ) = C(t) o_n(1). \end{aligned} \end{aligned}$$

Thus we have shown (6.33). Now, using (6.31)–(6.33) in (6.30) and returning to (6.29), we get that

$$\begin{aligned}&\varepsilon _n\int _{\Omega '} |\zeta |^2 |\mathrm {D}^{\mathrm {d}}(\Psi _n \circ {\widetilde{\chi }}_n^\perp )|^2 \, {\mathrm {d}{x}}\\&\quad \leqq \mathrm {Lip} (\Psi _n)^2 \varepsilon _n\int _{\Omega '} \eta _{n,t} |A^\mathrm {d}(\chi _n)|^2 \, {\mathrm {d}{x}} + Ct^4 + C(t) o_n(1) \, . \end{aligned}$$

By (6.7) and our assumption that $\Vert \Phi \Vert _{\mathrm {Ent}} \leqq 1$ we have that $\limsup _n \mathrm {Lip}(\Psi _n) \leqq 1$. Thus, letting $n \rightarrow \infty$ and then $t \rightarrow 0$ we infer that

$$\begin{aligned} \limsup _{n \rightarrow \infty } \varepsilon _n\int _{\Omega '} |\zeta |^2 |\mathrm {D}^{\mathrm {d}}(\Psi _n \circ {\widetilde{\chi }}_n^\perp )|^2 \, {\mathrm {d}{x}} \leqq \liminf _{n \rightarrow \infty } \varepsilon _n\int _{\Omega '} |A^\mathrm {d}( \chi _n)|^2 \, {\mathrm {d}{x}}, \end{aligned}$$

(6.34)

where we have used that $|\eta _{n,t}| \leqq 1$.

Step 10. (Conclusion.) By (6.10), (6.22), (6.23), (6.24), and (6.34) we conclude that

$$\begin{aligned} \langle \mathrm {div}({\widetilde{\Phi }} \circ \chi ^\perp ), \zeta \rangle \leqq \liminf _{n \rightarrow \infty } \frac{1}{2} \int _{\Omega '} \frac{1}{\varepsilon _n} W^\mathrm {d}(\chi _n) + \varepsilon _n|A^\mathrm {d}(\chi _n)|^2 \, {\mathrm {d}{x}}, \end{aligned}$$

i.e., (6.9) holds true as desired for all open $\Omega ' \subset \Omega$ and $\zeta \in C_c^\infty (\Omega ')$ with $\Vert \zeta \Vert _{L^\infty (\Omega ')} \leqq 1$. Passing to the supremum in $\zeta$, the left-hand side of the inequality becomes the total variation $|\mathrm {div}({\widetilde{\Phi }} \circ \chi ^\perp )|(\Omega ') = |\mathrm {div}(\Phi \circ \chi ^\perp )|(\Omega ')$. Then, considering partitions of $\Omega$ to pass to the supremum in $\Phi$ in the sense of measures, we get that

$$\begin{aligned} \bigvee _{\begin{array}{c} \Phi \in \mathrm {Ent}\\ \Vert \Phi \Vert _{\mathrm {Ent}} \leqq 1 \end{array}} |\mathrm {div}(\Phi \circ \chi ^\perp )|(\Omega ) \leqq \liminf _{n \rightarrow \infty } \frac{1}{2} \int _{\Omega } \frac{1}{\varepsilon _n} W^\mathrm {d}(\chi _n) + \varepsilon _n|A^\mathrm {d}(\chi _n)|^2 \, {\mathrm {d}{x}}. \end{aligned}$$

This is the claim (4.2) and concludes the proof of Theorem 4.1-ii).

Remark 6.2

To prove an analogous liminf inequality for the functionals $AG^{\mathrm {d}}_n$ defined by (2.23) in place of $H_n$, a similar proof can be used and several steps can be simplified substantially. In particular, the introduction of the approximations ${\widetilde{\Phi }}_n$ is not required. Moreover, there is no necessity to work with three different order parameters $\chi , {\widetilde{\chi }}, \overline{\chi }$ and to estimate errors that are created when replacing one with another.

For the functionals $AG^{\mathrm {d}}_n$, instead of (6.10), we have that

$$\begin{aligned} \langle \mathrm {div}({\widetilde{\Phi }} \circ \chi ^\perp ) , \zeta \rangle = \lim _n \int _{\Omega '} \zeta \mathrm {div}^{\mathrm {d}}({\widetilde{\Phi }} \circ \mathrm {D}^{\mathrm {d}}\varphi _n^\perp ) \, {\mathrm {d}{x}}, \end{aligned}$$

where we assume that $\mathrm {D}^{\mathrm {d}}\varphi _n \rightarrow \chi$ in $L^1_{\mathrm {loc}}$. Using that $\mathrm {curl}^{\mathrm {d}}(\mathrm {D}^{\mathrm {d}}\varphi _n )\equiv 0$, with the obvious simplifications Steps 6–6 yield that

$$\begin{aligned}&\int _{\Omega '} \zeta \mathrm {div}^{\mathrm {d}}({\widetilde{\Phi }} \circ \mathrm {D}^{\mathrm {d}}\varphi _n^\perp ) \, {\mathrm {d}{x}} \leqq \frac{1}{2} \int _{\Omega _n^{\mathrm {supp}}} \frac{1}{\varepsilon _n} q(\mathrm {D}^{\mathrm {d}}\varphi _n)^2 \, {\mathrm {d}{x}} \\&\quad + \frac{1}{2} \int _{\Omega '} \varepsilon _n|\zeta |^2 |\mathrm {div}^{\mathrm {d}}(\Psi \circ \mathrm {D}^{\mathrm {d}}\varphi _n^\perp )|^2 \, {\mathrm {d}{x}}. \end{aligned}$$

In place of Step 6, we get that $\int _{\Omega _n^{\mathrm {supp}}} q(\mathrm {D}^{\mathrm {d}}\varphi _n)^2 \, {\mathrm {d}{x}} \leqq \int _{\Omega '} W(\mathrm {D}^{\mathrm {d}}\varphi _n) \, {\mathrm {d}{x}}$, which is immediate, since $q(\xi )^2 = W(\xi )$. Then, performing a discrete integration by parts as in Step 6 we get that

$$\begin{aligned} \int _{\Omega '} \varepsilon _n|\zeta |^2 |\mathrm {div}^{\mathrm {d}}(\Psi \circ \mathrm {D}^{\mathrm {d}}\varphi _n^\perp )|^2 \, {\mathrm {d}{x}} \leqq \int _{\Omega '} \varepsilon _n|\zeta |^2 |\mathrm {D}^{\mathrm {d}}(\Psi \circ \mathrm {D}^{\mathrm {d}}\varphi _n^\perp )|^2 \, {\mathrm {d}{x}} + o_n(1) \end{aligned}$$

and using that $|\mathrm {D}^{\mathrm {d}}(\Psi \circ \mathrm {D}^{\mathrm {d}}\varphi _n^\perp )|^2 \leqq |\mathrm {D}^{\mathrm {d}}\mathrm {D}^{\mathrm {d}}\varphi _n|^2$ leads to the conclusion. The technical Steps 6–6 are not required.

In the next lemma we provide details about the approximate entropies ${\widetilde{\Phi }}_n$ that we have used above.

Lemma 6.3

Let $\Phi \in \mathrm {Ent}$, let $(\Psi , \alpha )$ be as in (3.2), (3.3), and let ${\widetilde{\Phi }}(\xi ) = \Phi (\xi ) - (1 - |\xi |^2) \Psi (\xi )$. Then, for n large enough, there exist functions ${\widetilde{\Phi }}_n \in C_c^\infty (\mathbb {R}^2 \setminus \{0\};\mathbb {R}^2)$, $\Psi _n \in C_c^\infty (\mathbb {R}^2 \setminus \{0\};\mathbb {R}^2)$, and $\alpha _n \in C_c^\infty (\mathbb {R}^2 \setminus \{0\})$ satisfying (6.5)–(6.8) above.

Proof

Following [26, Lemma 2.4, Formula (2.7)], let us define the scalar function $\phi \in C_c^\infty (\mathbb {R}^2 \setminus \{0\})$ by $\phi (\xi ) := \frac{1}{|\xi |^2} \Phi (\xi ) \cdot \xi$. Using (3.1), it can be checked that $\Phi$ is characterized by $\phi$ through¹⁴

$$\begin{aligned} \Phi (\xi ) = \phi (\xi ) \xi + (\nabla \phi (\xi ) \cdot \xi ^\perp ) \xi ^\perp . \end{aligned}$$

(6.35)

Before defining ${\widetilde{\Phi }}_n$, we first introduce an approximation $\Phi _n$ of $\Phi$ by using an approximate version of the above formula. We set $h_n(\xi ) := - \frac{1}{2} q_n(\xi )$. Then, as $n \rightarrow \infty$ we have that $\nabla h_n(\xi ) \rightarrow \xi$ and $\nabla ^2 h_n(\xi ) \rightarrow \mathrm {Id}$ locally uniformly in $\xi \in \mathbb {R}^2$. In fact, we even have¹⁵

$$\begin{aligned} h_n(\xi ) \rightarrow -\frac{1}{2} q(\xi ) = \frac{1}{2} (|\xi |^2 - 1) \quad \text {locally in } C^k \text { for every } k \in \mathbb {N}\end{aligned}$$

(6.36)

as $n \rightarrow \infty$, since the functions $s \mapsto \frac{2}{\sqrt{\delta _n}} \sin \big ( \frac{\sqrt{\delta _n}}{2}s \big )$ converge to the identity $s \mapsto s$ locally in $C^k$ for $k \in \mathbb {N}$. We define

$$\begin{aligned} \Phi _n := \phi \, \nabla h_n + \frac{|\nabla h_n|^2 (\nabla \phi \cdot \nabla ^\perp h_n) + \phi \, \nabla h_n \cdot (\nabla ^2 h_n \nabla ^\perp h_n)}{\nabla ^\perp h_n \cdot (\nabla ^2 h_n \nabla ^\perp h_n)} \nabla ^\perp h_n. \end{aligned}$$

(6.37)

For large enough n, this defines a function $\Phi _n \in C_c^\infty (\mathbb {R}^2 \setminus \{0\}; \mathbb {R}^2)$. Indeed, for large n, $\nabla ^2 h_n$ is positive definite and thus the denominator in the formula above can only be zero if $\nabla h_n = 0$. For large n this can only occur in a small neighborhood of the origin on which $\phi = 0$. From (6.35)–(6.37) we also get that $\Phi _n \rightarrow \Phi$ in $C^2$. The function $\Phi _n$ is defined in such a way that it satisfies an approximate version of condition (3.1), namely

$$\begin{aligned} \nabla h_n(\xi ) \cdot \big (\mathrm {D}\Phi _n(\xi ) \nabla ^\perp h_n(\xi )\big ) = 0 \quad \text {for all } \xi \in \mathbb {R}^2. \end{aligned}$$

(6.38)

To prove this, let us use the short-hand notation

$$\begin{aligned} f_n := \frac{ |\nabla h_n|^2 (\nabla \phi \cdot \nabla ^\perp h_n) + \phi \, \nabla h_n \cdot \big ( \nabla ^2 h_n \cdot \nabla ^\perp h_n \big ) }{\nabla ^\perp h_n \cdot \big ( \nabla ^2 h_n \cdot \nabla ^\perp h_n \big )}. \end{aligned}$$

We have that

$$\begin{aligned} \mathrm {D}\Phi _n = \nabla h_n \otimes \nabla \phi + \phi \nabla ^2 h_n + \nabla ^\perp h_n \otimes \nabla f_n + f_n \mathrm {D}\nabla ^\perp h_n. \end{aligned}$$

As a consequence,

$$\begin{aligned} \begin{aligned} \nabla h_n \cdot (\mathrm {D}\Phi _n \nabla ^\perp h_n)&= |\nabla h_n|^2 (\nabla \phi \cdot \nabla ^\perp h_n) + \phi \, \nabla h_n \cdot \big (\nabla ^2 h_n \nabla ^\perp h_n \big ) \\&\quad + f_n \nabla h_n \cdot \big (\mathrm {D}\nabla ^\perp h_n \nabla ^\perp h_n \big ). \end{aligned} \end{aligned}$$

Let for the moment $R \in SO(2)$ denote the rotation $x \mapsto x^\perp$. We observe that its inverse $R^{-1} = R^T$ is given by $-R$. Using that

$$\begin{aligned} \begin{aligned}&(\nabla h_n)^T \mathrm {D}\nabla ^\perp h_n = (\nabla h_n)^T \mathrm {D}(R \nabla h_n) = (\nabla h_n)^T R \nabla ^2 h_n = (R^T \nabla h_n)^T \nabla ^2 h_n \\&\qquad = - (\nabla ^\perp h_n)^T \nabla ^2 h_n, \end{aligned} \end{aligned}$$

we get that

$$\begin{aligned} \begin{aligned} \nabla h_n \cdot ( \mathrm {D}\Phi _n \nabla ^\perp h_n)&= |\nabla h_n|^2 (\nabla \phi \cdot \nabla ^\perp h_n) + \phi \, \nabla h_n \cdot \big (\nabla ^2 h_n \nabla ^\perp h_n \big ) \\&- f_n \nabla ^\perp h_n \cdot \big (\nabla ^2 h_n \nabla ^\perp h_n \big ) \\&= 0 \end{aligned} \end{aligned}$$

by the definition of $f_n$. This is (6.38).

Next we observe that (6.38) implies that

$$\begin{aligned} \mathrm {D}\Phi _n \nabla ^\perp h_n = \frac{\nabla ^\perp h_n \cdot (\mathrm {D}\Phi _n \nabla ^\perp h_n)}{|\nabla h_n|^2} \nabla ^\perp h_n. \end{aligned}$$

Using twice that $\mathrm {Id} = \frac{1}{|\nabla h_n|^2}(\nabla h_n \otimes \nabla h_n + \nabla ^\perp h_n \otimes \nabla ^\perp h_n)$, except in a small neighborhood of 0 in which $\Phi _n = 0$, the previous formula yields that

$$\begin{aligned} \begin{aligned} \mathrm {D}\Phi _n&= \frac{1}{|\nabla h_n|^2} \big ( \mathrm {D}\Phi _n \nabla h_n \otimes \nabla h_n + \mathrm {D}\Phi _n \nabla ^\perp h_n \otimes \nabla ^\perp h_n \big ) \\&= \frac{\nabla ^\perp h_n \cdot (\mathrm {D}\Phi _n \nabla ^\perp h_n)}{|\nabla h_n|^2} \mathrm {Id} \\&\quad + \frac{1}{|\nabla h_n|^2} \bigg ( \mathrm {D}\Phi _n \nabla h_n - \frac{\nabla ^\perp h_n \cdot (\mathrm {D}\Phi _n \nabla ^\perp h_n)}{|\nabla h_n|^2} \nabla h_n \bigg ) \otimes \nabla h_n. \end{aligned} \end{aligned}$$

Thus, we get an approximate version of (3.4), namely $\mathrm {D}\Phi _n + 2 \Psi _n \otimes \nabla h_n = \alpha _n \mathrm {Id}$, where we have set

$$\begin{aligned} \alpha _n := \frac{\nabla ^\perp h_n \cdot (\mathrm {D}\Phi _n \nabla ^\perp h_n)}{|\nabla h_n|^2} \quad \text {and} \quad \Psi _n := - \frac{1}{2 |\nabla h_n|^2} ( \mathrm {D}\Phi _n - \alpha _n \mathrm {Id}) \nabla h_n. \end{aligned}$$

Since $\Phi _n \rightarrow \Phi$ in $C^2$, by (6.36) and in view of (3.2), (3.3) we have that $\alpha _n \rightarrow \alpha$ and $\Psi _n \rightarrow \Psi$ in $C^2$. Now we define ${\widetilde{\Phi }}_n := \Phi _n - q_n \Psi _n$ and have that ${\widetilde{\Phi }}_n \rightarrow {\widetilde{\Phi }}$ in $C^2$ as claimed. This proves (6.5). Moreover, we have that

$$\begin{aligned} \mathrm {D}{\widetilde{\Phi }}_n = \alpha _n \mathrm {Id} - 2 \Psi _n \otimes h_n - \Psi _n \otimes \nabla q_n - q_n \mathrm {D}\Psi _n = \alpha _n \mathrm {Id} - q_n \mathrm {D}\Psi _n, \end{aligned}$$

which proves (6.6). Recalling Definition 3.2, (6.7) follows from (6.5). Finally, to obtain (6.8), let us observe that by the definition of $\phi$, $\Phi _n$, $\alpha _n$, $\Psi _n$, and ${\widetilde{\Phi }}_n$, we have that

$$\begin{aligned} \mathrm {supp}({\widetilde{\Phi }}_n) , \ \mathrm {supp}(\Psi _n) , \ \mathrm {supp}(\alpha _n) \subset \mathrm {supp}(\Phi _n) \subset \mathrm {supp}(\phi ) \subset \mathrm {supp}(\Phi ) \, , \end{aligned}$$

which is a compact set. $\square$

7 Proof of the Limsup Inequality

In this section we prove Theorem 4.1-iii). We recall that for the proof we need the additional assumption

$$\begin{aligned} \frac{\delta _n^{5/2}}{\lambda _n} \rightarrow 0 \quad \text {as } n \rightarrow \infty . \end{aligned}$$

We fix $\Omega \in \mathcal {A}_0$ and $\chi \in BV(\Omega ; \mathbb {R}^2)$ and we will prove that there exists a sequence $(\chi _n)_n \in L^1_{\mathrm {loc}}(\mathbb {R}^2; \mathbb {R}^2)$ with $\chi _n \rightarrow \chi$ in $L^1(\Omega ;\mathbb {R}^2)$ and

$$\begin{aligned} \limsup _{n \rightarrow \infty } H_n(\chi _n, \Omega ) \leqq H(\chi , \Omega ). \end{aligned}$$

(7.1)

If $H(\chi , \Omega ) = + \infty$ the statement is trivial. Hence, in what follows we will assume that $H(\chi , \Omega ) < + \infty$ and in particular that $\chi \in L^\infty (\Omega ; \mathbb {S}^1)$ and $\mathrm {curl}(\chi ) = 0$ in $\mathcal {D}'(\Omega )$. We recall that under our assumptions on $\Omega \in \mathcal {A}_0$, such a field $\chi$ admits a potential $\varphi \in BVG(\Omega )$ such that $\nabla \varphi = \chi$ (cf. [17, Lemma 3.4]). The potential $\varphi$ will be used in the construction of the recovery sequence below.

For a function $\chi$ with the properties listed above we will moreover show that there exists a sequence $(\chi _n)_n \in L^1_{\mathrm {loc}}(\mathbb {R}^2; \mathbb {R}^2)$ satisfying (7.1), such that additionally

$$\begin{aligned} \sup _n \Vert \chi _n \Vert _{L^\infty (\mathbb {R}^2)}< + \infty , \qquad \chi _n \rightarrow \chi \text { in } L^p(\Omega ; \mathbb {R}^2) \quad \text {for every } p < \infty . \end{aligned}$$

(7.2)

Relying on the idea that the functionals $H_n$ resemble a discrete version of Aviles–Giga functionals, we resort to the technique used in [46] to prove the limsup inequality for the classical Aviles–Giga functional. This technique has later been generalized by the same author in [47] to prove upper bounds for generic singular perturbation problems of the form

$$\begin{aligned} \frac{1}{\varepsilon _n} \int _{\Omega } F(\varepsilon _n\nabla ^2 \varphi (x), \nabla \varphi (x)) \, {\mathrm {d}{x}}. \end{aligned}$$

Led by the observation that the functionals $H_n$ resemble more closely the Aviles–Giga like energies $AG^\Delta _{\varepsilon _n}$ in (1.6), we will apply [47, Theorems 6.1, 6.2] to the sequence of functionals

$$\begin{aligned} \frac{1}{2} \int _{\Omega }\frac{1}{\varepsilon _n} W(\nabla \varphi ) + \varepsilon _n|\Delta \varphi |^2 \, {\mathrm {d}{x}}, \end{aligned}$$

(7.3)

i.e., to the case

$$\begin{aligned} F(A,b) = \frac{1}{2} \big ( W(b) + |\mathrm {tr} (A)|^2 \big ) \quad \text {for } A \in \mathbb {R}^{2 {\times }2}, \ b \in \mathbb {R}^2. \end{aligned}$$

(7.4)

Before proving (7.1), we recall that the technique proposed in [47] uses a sequence of mollifications of $\varphi$ to obtain a candidate for the recovery sequence. This leads to an asymptotic upper bound for the functionals in (7.3) which depends on the choice of the mollifier. Subsequently, the limsup inequality is obtained by optimizing the upper bound over all admissible mollifiers.

To define a mollification of $\varphi$ on $\Omega$ we first extend it to the whole $\mathbb {R}^2$. Since $\Omega$ is a BVG domain, by Proposition 2.1 we can find a compactly supported function $\overline{\varphi }\in BVG(\mathbb {R}^2)$ such that $\overline{\varphi }= \varphi$ a.e. in $\Omega$ and $|\mathrm {D}\nabla \overline{\varphi }|(\partial \Omega ) = 0$.

We define a sequence $\varphi ^\varepsilon$ by convolving $\overline{\varphi }$ with suitable kernels. Following [47], we introduce the class $\mathcal {V}(\Omega )$ consisting of mollifiers $\eta \in C^3_c(\mathbb {R}^2 {\times }\mathbb {R}^2; \mathbb {R})$ satisfying that

$$\begin{aligned} \int _{\mathbb {R}^2} \eta (z,x) \, {\mathrm {d}{z}} = 1 \quad \text {for all } x \in \Omega \, . \end{aligned}$$

(7.5)

Remark 7.1

In [47] the author only requires $C^2$ regularity for the mollifiers. We remark that the proofs of [47, Theorem 6.1, Theorem 6.2] also work under this stronger regularity assumption on the convolution kernels.

Let us fix an arbitrary mollifier $\eta \in \mathcal {V}(\Omega )$ and let us define

$$\begin{aligned} \varphi ^{\varepsilon }(x)&:= \frac{1}{\varepsilon ^2} \int _{\mathbb {R}^2}\eta \big ( \tfrac{y-x}{\varepsilon } , x\big ) \overline{\varphi }(y) \, {\mathrm {d}{y}} \nonumber \\&= \int _{\mathbb {R}^2}\eta ( z , x ) \overline{\varphi }( x + \varepsilon z) \, {\mathrm {d}{z}} \quad \text {for } x \in \mathbb {R}^2 . \end{aligned}$$

(7.6)

Evaluating the sequence of functionals in (7.3) on the functions $\varphi ^{\varepsilon _n}$, we obtain a first asymptotic upper bound. More precisely, by [47, Theorem 6.1] we have that

$$\begin{aligned} \lim _{n \rightarrow \infty } \int _{\Omega } \frac{1}{\varepsilon _n} W(\nabla \varphi ^{\varepsilon _n}) + \varepsilon _n|\Delta \varphi ^{\varepsilon _n}|^2 \, {\mathrm {d}{x}} = Y[\eta ](\varphi ), \end{aligned}$$

(7.7)

where an explicit formula for $Y[\eta ](\varphi )$ is given in [47, Formula (6.4)]. The precise expression of $Y[\eta ](\varphi )$ is not relevant for our purposes. It is however important to derive the expression obtained when we optimize $Y[\eta ](\varphi )$ with respect to $\eta \in \mathcal {V}(\Omega )$.

Proposition 7.2

The following equality holds true:

$$\begin{aligned} \inf _{\eta \in \mathcal {V}(\Omega )} Y[\eta ](\varphi ) = \frac{1}{6} \int _{J_\chi } |[\chi ]|^3 \, {\mathrm {d}{\mathcal {H}}}^1. \end{aligned}$$

Proof

We recall that [47, Theorem 6.2] gives

$$\begin{aligned} \inf _{\eta \in \mathcal {V}(\Omega )} Y[\eta ](\varphi ) = \int _{J_{\nabla \varphi }} \sigma \big (\nabla \varphi ^+(x), \nabla \varphi ^-(x), \nu _{\nabla \varphi }(x) \big ) \, {\mathrm {d}{\mathcal {H}}}^1(x) , \end{aligned}$$

where the surface density $\sigma$ is obtained by optimizing the energy for a transition from $\nabla \varphi ^-(x)$ to $\nabla \varphi ^+(x)$ over one-dimensional profiles and is given by

$$\begin{aligned} \begin{aligned}&\sigma (a,b,\nu )\\&\quad := \inf _{\gamma } \Big \{ \int _{-\infty }^{+\infty } F\big ( - \gamma '(t) \, \nu \otimes \nu , \gamma (t) \, \nu + b\big ) \, {\mathrm {d}{t}} \ : \ \gamma \in C^1(\mathbb {R}), \text { there exists } L > 0 \\&\qquad \text { s.t.\ for } t \geqq L \text { we have } \gamma (-t) = d{} \text { and } \gamma (t) = 0 \Big \} \end{aligned} \end{aligned}$$

for every $a, b \in \mathbb {R}^2$ and $\nu \in \mathbb {S}^1$ such that $(a-b) = d{} \, \nu$ for some $d{} \in \mathbb {R}$. This exhaustively defines the energy for the triple $\big (\nabla \varphi ^+(x), \nabla \varphi ^-(x), \nu _{\nabla \varphi } (x) \big )$ for every $x \in J_{\nabla \varphi }$, cf. Section 2.3.

We claim that for all $a,b \in \mathbb {S}^1$, $a \ne b$, and $\nu \in \mathbb {S}^1$ with $(a-b) = d{} \, \nu$, $d{} \in \mathbb {R}$, we have that

$$\begin{aligned} \sigma \big ( a, b, \nu \big ) = \frac{1}{6} |a-b|^3. \end{aligned}$$

(7.8)

In particular, since $\nabla \varphi ^\pm = \chi ^\pm \in \mathbb {S}^1$ a.e., we obtain that $\sigma \big ( \nabla \varphi ^+, \nabla \varphi ^-, \nu _{\nabla \varphi } \big ) = \frac{1}{6} | [\chi ]|^3$ $\mathcal {H}^1$-a.e. on $J_{\nabla \varphi } = J_{\chi }$. This will conclude the proof.

To prove (7.8), let us consider any admissible profile $\gamma$ in the infimum problem defining $\sigma (a,b,\nu )$. Using the definition of F in (7.4) together with $|\mathrm {tr} (\nu \otimes \nu )| = |\nu |^2 = 1$ and writing $\gamma (t) \nu + b = (b \cdot \nu ^\perp )\nu ^\perp + (\gamma (t) + b \cdot \nu )\nu$, we get that

$$\begin{aligned} \begin{aligned}&\int _{-\infty }^{+\infty } F\big ( - \gamma '(t) \, \nu \otimes \nu , \gamma (t) \, \nu + b\big ) \, {\mathrm {d}{t}} \\&\quad = \frac{1}{2} \int _{-\infty }^{+ \infty } \big ( 1 - |b \cdot \nu ^\perp |^2 - |\gamma (t) + b \cdot \nu |^2 \big )^2 + |\gamma '(t)|^2 \, {\mathrm {d}{t}}. \end{aligned} \end{aligned}$$

Next, note that our assumptions $a,b,\nu \in \mathbb {S}^1$, $a \ne b$, and $(a-b) = d{} \, \nu$ imply that $a \cdot \nu = - b \cdot \nu = \frac{d{}}{2}$ and $1 - |b \cdot \nu ^\perp |^2 = \frac{|d{}|^2}{4}$. In conclusion we obtain that

$$\begin{aligned} \begin{aligned}&\int _{-\infty }^{+\infty } F\big ( - \gamma '(t) \, \nu \otimes \nu , \gamma (t) \, \nu + b\big ) \, {\mathrm {d}{t}} \\&\quad = \frac{1}{2} \int _{-\infty }^{+ \infty } \Big ( \tfrac{|d{}|^2}{4} - \big | \gamma (t) - \tfrac{d{}}{2} \big |^2 \Big )^2 + |\gamma '(t)|^2 \, {\mathrm {d}{t}} \\&\quad = \frac{1}{2} \int _{-\infty }^{+ \infty } \tfrac{|d{}|^4}{16} \Big ( 1 - \big | {\widetilde{\gamma }} \big (\tfrac{|d{}|}{2} t \big ) \big |^2 \Big )^2 + \tfrac{|d{}|^4}{16} \big |{\widetilde{\gamma }}' \big (\tfrac{|d{}|}{2}t \big ) \big |^2 \, {\mathrm {d}{t}}, \end{aligned} \end{aligned}$$

where we have put ${\widetilde{\gamma }}(t) := \frac{2}{d{}} \big ( \gamma (\frac{2}{|d{}|}t) - \frac{d{}}{2} \big )$. Using the change of variables $s = \frac{|d{}|}{2} t$ we infer that

$$\begin{aligned} \int _{-\infty }^{+\infty } F\big ( - \gamma '(t) \, \nu \otimes \nu , \gamma (t) \, \nu + b\big ) \, {\mathrm {d}{t}} = \frac{|d{}|^3}{16} \int _{-\infty }^{+ \infty } \big ( 1 - | {\widetilde{\gamma }}(s)|^2 \big )^2 + |{\widetilde{\gamma }}'(s)|^2 \, {\mathrm {d}{s}}. \end{aligned}$$

Note that ${\widetilde{\gamma }}(s) = -1$ for s large enough and ${\widetilde{\gamma }}(s) = 1$ for s small enough. Thus, up to the multiplicative factor $\frac{|d{}|^3}{16}$, the infimum problem that defines $\sigma (a,b,\nu )$ coincides with the infimum problem for the optimal profile of the Modica-Mortola functional, cf. for example [14, Chapter 6]. In conclusion, $\sigma (a,b,\nu ) = \frac{|d{}|^3}{16} \, 2 \big | \int _{-1}^1 (1-s^2) \, {\mathrm {d}{s}} \big | = \frac{|d{}|^3}{6}$. Since $|a-b| = |d{}|$, we conclude (7.8). $\square$

As a consequence, to prove Theorem 4.1-iii) we now only need to construct a sequence of spin fields $u_n \in \mathcal {PC}_{\lambda _n}(\mathbb {S}^1)$ such that their associated chirality variables $\chi _n$ satisfy (7.2) and

$$\begin{aligned} \limsup _n H_n(\chi _n, \Omega ) \leqq Y[\eta ](\chi ). \end{aligned}$$

(7.9)

Indeed, in view of Proposition 7.2 and Corollary 3.8, the existence of a recovery sequence satisfying (7.1), (7.2) is then obtained by a diagonal argument (cf. also [47, Section 5]). To find such a sequence $(u_n)_n$, we first discretize the functions $\varphi ^{\varepsilon _n}$ defined by (7.6) on the lattice $\lambda _n\mathbb {Z}^2$. Specifically, we define $\varphi _n \in \mathcal {PC}_{\lambda _n}(\mathbb {R})$ by

$$\begin{aligned} \varphi _n^{i,j} := \varphi ^{\varepsilon _n}(\lambda _ni , \lambda _nj). \end{aligned}$$

In the next proposition we prove that the Aviles–Giga-like functionals in (7.7) are the same as their discrete counterparts evaluated on $\varphi _n$, up to an error that vanishes when $n \rightarrow \infty$.

Proposition 7.3

We have that

$$\begin{aligned} \frac{1}{2} \int _{\Omega } \frac{1}{\varepsilon _n} W(\nabla \varphi ^{\varepsilon _n}) + \varepsilon _n|\Delta \varphi ^{\varepsilon _n}|^2 \, {\mathrm {d}{x}} = \frac{1}{2} \int _\Omega \frac{1}{\varepsilon _n} W(\mathrm {D}^{\mathrm {d}}\varphi _n) + \varepsilon _n|\Delta _{\mathrm {s}}^{\mathrm {d}}\varphi _n|^2 \, {\mathrm {d}{x}} + o_n(1), \end{aligned}$$

where $\Delta _{\mathrm {s}}^{\mathrm {d}}\varphi _n$ is defined by (2.3).

Proof

Step 1. ($L^\infty$-bounds on derivatives of $\varphi ^{\varepsilon _n}$.) We claim that there exists a constant $C > 0$ such that

$$\begin{aligned} \big | \nabla \varphi ^{\varepsilon _n}(x) \big |&\leqq C, \quad&\big |\mathrm {D}^{\mathrm {d}}\varphi _n(x) \big |&\leqq C, \end{aligned}$$

(7.10)

$$\begin{aligned} |\nabla ^2 \varphi ^{\varepsilon _n}(x)|&\leqq \frac{C}{\varepsilon _n}, \quad&|\mathrm {D}^{\mathrm {d}}\mathrm {D}^{\mathrm {d}}\varphi _n(x)|&\leqq \frac{C}{\varepsilon _n}, \end{aligned}$$

(7.11)

$$\begin{aligned} |\nabla ^3 \varphi ^{\varepsilon _n}(x)|&\leqq \frac{C}{\varepsilon _n^2}, \end{aligned}$$

(7.12)

for every $x \in \mathbb {R}^2$ and every n. From the very definition of $\varphi ^{\varepsilon _n}$ in (7.6) and by integrating by parts we get

$$\begin{aligned} \begin{aligned} \partial _k \varphi ^{\varepsilon _n}(x)&= \frac{1}{\varepsilon _n^2} \int _{\mathbb {R}^2} \eta (\tfrac{y-x}{\varepsilon _n},x) \partial _k \overline{\varphi }(y) \, {\mathrm {d}{y}} + \frac{1}{\varepsilon _n^2} \int _{\mathbb {R}^2} \partial _{x_k} \eta (\tfrac{y-x}{\varepsilon _n},x) \overline{\varphi }(y) \, {\mathrm {d}{y}} \, , \end{aligned} \end{aligned}$$

(7.13)

$$\begin{aligned} \begin{aligned} \partial _{hk} \varphi ^{\varepsilon _n}(x)&= - \frac{1}{\varepsilon _n^3} \int _{\mathbb {R}^2} \partial _{z_h} \eta (\tfrac{y-x}{\varepsilon _n},x) \partial _{k} \overline{\varphi }(y) \, {\mathrm {d}{y}} + \frac{1}{\varepsilon _n^2} \int _{\mathbb {R}^2} \partial _{x_h x_k} \eta (\tfrac{y-x}{\varepsilon _n},x)\overline{\varphi }(y) \, {\mathrm {d}{y}} \\&\quad + \frac{1}{\varepsilon _n^2} \int _{\mathbb {R}^2} \partial _{x_h} \eta (\tfrac{y-x}{\varepsilon _n},x) \partial _{k} \overline{\varphi }(y) \, {\mathrm {d}{y}} + \frac{1}{\varepsilon _n^2} \int _{\mathbb {R}^2}\partial _{x_k} \eta (\tfrac{y-x}{\varepsilon _n},x) \partial _{h} \overline{\varphi }(y) \, {\mathrm {d}{y}}, \end{aligned}\nonumber \\ \end{aligned}$$

(7.14)

and

$$\begin{aligned} \begin{aligned}&\partial _{hk \ell } \varphi ^{\varepsilon _n}(x) = \frac{1}{\varepsilon _n^4} \int _{\mathbb {R}^2}\partial _{z_h z_\ell }\eta (\tfrac{y-x}{\varepsilon _n},x) \partial _{k} \overline{\varphi }(y) \, {\mathrm {d}{y}} - \frac{1}{\varepsilon _n^3} \int _{\mathbb {R}^2}\partial _{z_h x_\ell }\eta (\tfrac{y-x}{\varepsilon _n},x) \partial _{k} \overline{\varphi }(y) \, {\mathrm {d}{y}} \\&\quad - \frac{1}{\varepsilon _n^3} \int _{\mathbb {R}^2}\partial _{z_\ell x_h x_k} \eta (\tfrac{y-x}{\varepsilon _n},x)\overline{\varphi }(y) \, {\mathrm {d}{y}} + \frac{1}{\varepsilon _n^2} \int _{\mathbb {R}^2}\partial _{x_h x_k x_\ell } \eta (\tfrac{y-x}{\varepsilon _n},x)\overline{\varphi }(y) \, {\mathrm {d}{y}}\\&\quad - \frac{1}{\varepsilon _n^3} \int _{\mathbb {R}^2}\partial _{z_\ell x_h} \eta (\tfrac{y-x}{\varepsilon _n},x) \partial _{k} \overline{\varphi }(y) \, {\mathrm {d}{y}} + \frac{1}{\varepsilon _n^2} \int _{\mathbb {R}^2}\partial _{x_h x_\ell } \eta (\tfrac{y-x}{\varepsilon _n},x) \partial _{k} \overline{\varphi }(y) \, {\mathrm {d}{y}} \\&\quad - \frac{1}{\varepsilon _n^3} \int _{\mathbb {R}^2} \partial _{z_\ell x_k} \eta (\tfrac{y-x}{\varepsilon _n},x) \partial _{h} \overline{\varphi }(y) \, {\mathrm {d}{y}} + \frac{1}{\varepsilon _n^2} \int _{\mathbb {R}^2} \partial _{x_k x_\ell } \eta (\tfrac{y-x}{\varepsilon _n},x) \partial _{h} \overline{\varphi }(y) \, {\mathrm {d}{y}}, \end{aligned} \end{aligned}$$

(7.15)

where $\partial _{z_h} \eta (z,x)$ and $\partial _{x_h} \eta (z,x)$ denote the derivative with respect to the h-th variable in the first and second group of variables of $\eta (z,x)$ respectively.

By the assumptions on $\eta$, the function $y \mapsto \eta \big (\tfrac{y - x}{\varepsilon _n},x\big )$ is supported on a ball $B_{R \varepsilon _n}(x)$ for a suitable $R > 0$ (independent of n and x). Together with the condition $\overline{\varphi }\in W^{1,\infty }(\mathbb {R}^2)$, (7.13)–(7.15) yield the first inequalities in (7.10)–(7.12).

Next, we observe that, as a consequence,

$$\begin{aligned}&|\partial ^{\mathrm {d}}_1 \varphi _n^{i,j}| = \Big | \frac{\varphi ^{\varepsilon _n}(\lambda _n(i+1),\lambda _nj) - \varphi ^{\varepsilon _n}(\lambda _ni,\lambda _nj)}{\lambda _n} \Big | \\&\quad \leqq \int _0^1 \big | \partial _1 \varphi ^{\varepsilon _n}(\lambda _n(i+t),\lambda _nj) \big | \, {\mathrm {d}{t}} \leqq C. \end{aligned}$$

With analogous computations for $|\partial ^{\mathrm {d}}_2 \varphi _n^{i,j}|$ we conclude the second inequality in (7.10).

In a similar way we also get that

$$\begin{aligned} |\partial ^{\mathrm {d}}_{kh} \varphi _n^{i,j}| \leqq \int _0^1 \int _0^1 \big | \partial _{kh} \varphi ^{\varepsilon _n} \big (\lambda _n(i,j) + \lambda _nt e_k + \lambda _ns e_h \big ) \big | \, {\mathrm {d}{s}} \, {\mathrm {d}{t}} \leqq \frac{C}{\varepsilon _n} \end{aligned}$$

(7.16)

and thereby the second inequality in (7.11).

Step 2. ($L^1$-bounds on derivatives of order 2.) We prove that

$$\begin{aligned}&\int _{\mathbb {R}^2} \sup _{B_{\sqrt{5} \lambda _n}(x)} \! \! |\nabla ^2 \varphi ^{\varepsilon _n}| \, {\mathrm {d}{x}} \leqq C, \end{aligned}$$

(7.17)

$$\begin{aligned}&\Vert \mathrm {D}^{\mathrm {d}}\mathrm {D}^{\mathrm {d}}\varphi _n \Vert _{L^1(\mathbb {R}^2)} \leqq C. \end{aligned}$$

(7.18)

Recalling that $\nabla \overline{\varphi }\in BV(\mathbb {R}^2;\mathbb {R}^2)$, we can integrate by parts in (7.14) and obtain that

$$\begin{aligned} \begin{aligned} \partial _{hk} \varphi ^{\varepsilon _n}(x)&= \frac{1}{\varepsilon _n^2} \int _{\mathbb {R}^2}\eta (\tfrac{y-x}{\varepsilon _n},x) \, {\mathrm {d}{\mathrm {D}}}_{h}\partial _{k} \overline{\varphi }(y) + \frac{1}{\varepsilon _n^2} \int _{\mathbb {R}^2} \partial _{x_h x_k} \eta (\tfrac{y-x}{\varepsilon _n},x)\overline{\varphi }(y) \, {\mathrm {d}{y}} \\&\quad + \frac{1}{\varepsilon _n^2} \int _{\mathbb {R}^2} \partial _{x_h} \eta (\tfrac{y-x}{\varepsilon _n},x) \partial _{k} \overline{\varphi }(y) \, {\mathrm {d}{y}} + \frac{1}{\varepsilon _n^2} \int _{\mathbb {R}^2}\partial _{x_k} \eta (\tfrac{y-x}{\varepsilon _n},x) \partial _{h} \overline{\varphi }(y) \, {\mathrm {d}{y}}, \end{aligned} \end{aligned}$$

where we let $\mathrm {D}_h \partial _{k} \overline{\varphi }$ denote the h-th component of the distributional derivative of $\partial _{k} \overline{\varphi }$. Since the function $y \mapsto \eta \big (\tfrac{y - x}{\varepsilon _n},x \big )$ is supported on a ball $B_{R \varepsilon _n}(x)$, we observe that for every $x \in \mathbb {R}^2$

$$\begin{aligned} |\nabla ^2 \varphi ^{\varepsilon _n}(x)| \leqq \Vert \eta \Vert _{\infty } \frac{1}{\varepsilon _n^2} |\mathrm {D}\nabla \overline{\varphi }|\big (B_{R \varepsilon _n}(x)\big ) + C \end{aligned}$$

and therefore

$$\begin{aligned} \sup _{ B_{\sqrt{5} \lambda _n}(x)} \! \! |\nabla ^2 \varphi ^{\varepsilon _n}| \leqq C \Big ( 1 + \frac{1}{\varepsilon _n^2} |\mathrm {D}\nabla \overline{\varphi }|\big ( B_{\sqrt{5} \lambda _n+ R \varepsilon _n}(x) \big ) \Big ). \end{aligned}$$

Since $\overline{\varphi }$ is compactly supported in $\mathbb {R}^2$, all $\varphi ^{\varepsilon _n}$ are supported in a common bounded set K. As a consequence, we get that

$$\begin{aligned} \begin{aligned} \int _{\mathbb {R}^2} \sup _{B_{\sqrt{5} \lambda _n}(x)} \! \! |\nabla ^2 \varphi ^{\varepsilon _n}| \, {\mathrm {d}{x}}&\leqq \int _{K + B_{\sqrt{5} \lambda _n}} C \Big ( 1 + \frac{1}{\varepsilon _n^2} |\mathrm {D}\nabla \overline{\varphi }|\big ( B_{\sqrt{5} \lambda _n+ R \varepsilon _n}(x) \big ) \Big ) \, {\mathrm {d}{x}} \\&\leqq C + \frac{1}{\varepsilon _n^2} \int _{\mathbb {R}^2} |\mathrm {D}\nabla \overline{\varphi }|\big ( B_{\sqrt{5} \lambda _n+ R \varepsilon _n}(x) \big ) \, {\mathrm {d}{x}}. \end{aligned} \end{aligned}$$

By Fubini we have that

$$\begin{aligned}&\frac{1}{\varepsilon _n^2} \int _{\mathbb {R}^2} |\mathrm {D}\nabla \overline{\varphi }|\big ( B_{\sqrt{5} \lambda _n+ R \varepsilon _n}(x) \big ) \, {\mathrm {d}{x}} \\&\quad = \frac{1}{\varepsilon _n^2} \int _{\mathbb {R}^2} \mathcal {L}^2 \big ( B_{\sqrt{5} \lambda _n+ R \varepsilon _n}(x') \big ) \, {\mathrm {d}{|}}\mathrm {D}\nabla \overline{\varphi }|(x') \leqq C |\mathrm {D}\nabla \overline{\varphi }|(\mathbb {R}^2), \end{aligned}$$

where we have used that $\lambda _n\ll \varepsilon _n$ as $n \rightarrow \infty$ by (2.13). This concludes the proof of (7.17). To prove (7.18) it only remains to observe that with the estimate (7.16) we get that, for $x \in Q_{\lambda _n}(i,j)$

$$\begin{aligned} |\partial ^{\mathrm {d}}_{kh} \varphi _n(x)| \leqq \int _0^1 \int _0^1 \big | \partial _{kh} \varphi ^{\varepsilon _n} \big (\lambda _n(i,j) + \lambda _nt e_k + \lambda _ns e_h \big ) \big | \, {\mathrm {d}{s}} \, {\mathrm {d}{t}} \leqq \sup _{B_{\sqrt{5} \lambda _n}(x)} |\nabla ^2 \varphi ^{\varepsilon _n}|. \end{aligned}$$

Step 3. (Estimates on the error in the potential part.) We show that

$$\begin{aligned} \frac{1}{\varepsilon _n} \int _{\Omega } \big | W(\nabla \varphi ^{\varepsilon _n}(x)) - W(\mathrm {D}^{\mathrm {d}}\varphi _n(x)) \big | \, {\mathrm {d}{x}} \rightarrow 0. \end{aligned}$$

We start by observing that for every $x \in Q_{\lambda _n}(i,j)$

$$\begin{aligned} \begin{aligned} \big |\partial _1 \varphi ^{\varepsilon _n}(x) - \partial _1^{\mathrm {d}} \varphi _n(x) \big |&= \big |\partial _1 \varphi ^{\varepsilon _n}(x) - \partial _1^{\mathrm {d}} \varphi _n^{i,j} \big | \\&\leqq \int _0^1 \big | \partial _1 \varphi ^{\varepsilon _n}(x) - \partial _1 \varphi ^{\varepsilon _n}(\lambda _n(i+t), \lambda _nj) \big | \, {\mathrm {d}{t}} \\&\leqq \sup _{ B_{\sqrt{2} \lambda _n}(x)} |\nabla ^2 \varphi ^{\varepsilon _n}| \sqrt{2} \lambda _n, \end{aligned} \end{aligned}$$

(7.19)

a similar computation being true for the discrete partial derivatives in the direction of $e_2$. By (7.10) and since W is locally Lipschitz, there exists a constant L independent of n and x, such that

$$\begin{aligned}&\big | W(\nabla \varphi ^{\varepsilon _n}(x)) - W(\mathrm {D}^{\mathrm {d}}\varphi _n(x)) \big | \leqq L \big |\nabla \varphi ^{\varepsilon _n}(x) - \mathrm {D}^{\mathrm {d}}\varphi _n(x) \big | \\&\quad \leqq L \sup _{ B_{\sqrt{2} \lambda _n}(x)}|\nabla ^2 \varphi ^{\varepsilon _n}| \sqrt{2} \lambda _n\, . \end{aligned}$$

By (7.17) and using (2.13) we get that

$$\begin{aligned} \frac{1}{\varepsilon _n} \int _{\Omega } \big | W(\nabla \varphi ^{\varepsilon _n}(x)) - W(\mathrm {D}^{\mathrm {d}}\varphi _n(x)) \big | \, {\mathrm {d}{x}} \leqq \sqrt{2}L C \frac{\lambda _n}{\varepsilon _n} = C \sqrt{\delta _n} \rightarrow 0. \end{aligned}$$

Step 4. (Estimates on the error in the derivative part.) We show that

$$\begin{aligned} \varepsilon _n\int _{\Omega } \big | |\Delta \varphi ^{\varepsilon _n}|^2 - | \Delta _{\mathrm {s}}^{\mathrm {d}}\varphi _n |^2 \big | \, {\mathrm {d}{x}} \rightarrow 0. \end{aligned}$$

To this end, we observe again that for $x \in Q_{\lambda _n}(i,j)$

$$\begin{aligned}&\partial _{11}^{\mathrm {d}} \varphi _n^{i-1,j} - \partial _{11} \varphi ^{\varepsilon _n} (x)\\&\quad = \int _0^1 \int _0^1 \partial _{11}\varphi ^{\varepsilon _n}(\lambda _n(i-1+s+t),\lambda _nj) - \partial _{11} \varphi ^{\varepsilon _n}(x) \, {\mathrm {d}{s}} \, {\mathrm {d}{t}} \end{aligned}$$

and thus, noting that $|x - (\lambda _n(i-1+s+t), \lambda _nj)| \leqq \sqrt{5} \lambda _n$, we conclude that

$$\begin{aligned} | \partial _{11}^{\mathrm {d}} \varphi _n^{i-1,j} - \partial _{11} \varphi ^{\varepsilon _n} (x) | \leqq \sqrt{5}\lambda _n\Vert \nabla ^3 \varphi ^{\varepsilon _n} \Vert _{L^\infty (\mathbb {R}^2)} \leqq C \frac{\lambda _n}{\varepsilon _n^2}, \end{aligned}$$

where we have used (7.12). Since the same estimate holds true for $| \partial _{22}^{\mathrm {d}} \varphi _n^{i,j-1} - \partial _{22} \varphi ^{\varepsilon _n} (x) |$, we infer that

$$\begin{aligned} \begin{aligned}&\varepsilon _n\int _{\mathbb {R}^2} \big | |\Delta \varphi ^{\varepsilon _n}|^2 - | \Delta _{\mathrm {s}}^{\mathrm {d}}\varphi _n |^2 \big | \, {\mathrm {d}{x}} \\&\quad = \varepsilon _n\int _{\mathbb {R}^2} \big | \Delta \varphi ^{\varepsilon _n} - \Delta _{\mathrm {s}}^{\mathrm {d}}\varphi _n \big | \big | \Delta \varphi ^{\varepsilon _n} + \Delta _{\mathrm {s}}^{\mathrm {d}}\varphi _n \big | \, {\mathrm {d}{x}} \\&\quad \leqq C \frac{\lambda _n}{\varepsilon _n} \Big ( \Vert \nabla ^2 \varphi ^{\varepsilon _n} \Vert _{L^1(\mathbb {R}^2)} + \Vert \mathrm {D}^{\mathrm {d}}\mathrm {D}^{\mathrm {d}}\varphi _n \Vert _{L^1(\mathbb {R}^2)} \Big ) \leqq C \sqrt{\delta _n} \rightarrow 0, \end{aligned} \end{aligned}$$

where we have used (7.17)–(7.18) and (2.13). This concludes the proof. $\square$

Remark 7.4

In view of (7.7), Proposition 7.3 yields

$$\begin{aligned} \lim _{n \rightarrow \infty } \frac{1}{2} \int _\Omega \frac{1}{\varepsilon _n} W(\mathrm {D}^{\mathrm {d}}\varphi _n) + \varepsilon _n|\Delta _{\mathrm {s}}^{\mathrm {d}}\varphi _n|^2 \, {\mathrm {d}{x}} = Y[\eta ](\varphi ). \end{aligned}$$

(7.20)

Together with $\varphi _n \rightarrow \varphi$ in $W^{1,1}(\Omega )$ (see the proof of Proposition 7.5 below) this allows us to prove the limsup inequality on the space of $\varphi \in BVG(\Omega )$ such that $|\nabla \varphi | = 1$ a.e. for the discrete functionals in (7.20). In a similar fashion it is possible to prove the same limsup inequality for the discrete Aviles–Giga functionals defined in (2.23). Note that both results hold without assuming the additional scaling assumption $\frac{\delta _n^{5/2}}{\lambda _n} \rightarrow 0$ and instead require merely that $\frac{\lambda _n}{\varepsilon _n} \rightarrow 0$ as $n \rightarrow \infty$.

Using the discrete functions $\varphi _n$, we can now define the sequence $\chi _n$. To this end it is convenient to introduce the spin fields $u_n \in \mathcal {PC}_{\lambda _n}(\mathbb {S}^1)$ by

$$\begin{aligned} u_n^{i,j} := \Big ( \cos \big ( \tfrac{\sqrt{\delta _n}}{\lambda _n} \varphi _n^{i,j} \big ), \, \sin \big ( \tfrac{\sqrt{\delta _n}}{\lambda _n} \varphi _n^{i,j} \big ) \Big ). \end{aligned}$$

We then define $\chi _n := \chi (u_n)$ through (2.8) as the chirality variable associated to $u_n$. Moreover, let us again use the notation ${\widetilde{\chi }}_n := {\widetilde{\chi }}(u_n)$ for the order parameters defined as well in (2.8), and $\overline{\chi }_n$ for the auxiliary variables defined by (2.19).

Note that the construction of $u_n$ is done in such a way that

$$\begin{aligned} \overline{\chi }_n = \mathrm {D}^{\mathrm {d}}\varphi _n \quad \text {for all { n} large enough.} \end{aligned}$$

(7.21)

Indeed, by (7.10) we have that $\sqrt{\delta _n} |\partial ^{\mathrm {d}}_1 \varphi _n| < \pi$ for all n large enough. Thus, evaluating (2.6) and using standard trigonometric identities we get that

$$\begin{aligned} (\theta ^{\mathrm {hor}}(u_n))^{i,j} = \mathrm {sign} \big (\sin (\sqrt{\delta _n} \, \partial ^{\mathrm {d}}_1 \varphi _n^{i,j}) \big ) \arccos \big ( \cos ( \sqrt{\delta _n} \, \partial ^{\mathrm {d}}_1 \varphi _n^{i,j}) \big ) = \sqrt{\delta _n} \, \partial ^{\mathrm {d}}_1 \varphi _n^{i,j} \end{aligned}$$

for all i, j. Analogously, $(\theta ^{\mathrm {ver}}(u_n))^{i,j} = \sqrt{\delta _n} \, \partial ^{\mathrm {d}}_2 \varphi _n^{i,j}$. Then, in view of (2.19) we obtain (7.21).

Let us prove that the sequence $(\chi _n)_n$ satisfies the conditions in (7.2).

Proposition 7.5

There exists a constant $C > 0$ such that

$$\begin{aligned} \Vert \chi _n \Vert _{L^\infty (\mathbb {R}^2)} \leqq C \quad \text {and} \quad \Vert \overline{\chi }_n \Vert _{L^\infty (\mathbb {R}^2)} \leqq C \end{aligned}$$

(7.22)

for all n. Moreover, $\chi _n \rightarrow \chi$ in $L^p(\Omega ;\mathbb {R}^2)$ for all $p < \infty$.

Proof

From (7.10) and (7.21) we immediately get boundedness of $(\overline{\chi }_n)_n$ in $L^\infty$. Writing $\chi _n$ in terms of $\overline{\chi }_n$ and using that $|\sin (s)| \leqq |s|$ we have that $|\chi _n| \leqq |\overline{\chi }_n|$, which concludes the proof of (7.22).

To show that $\chi _n \rightarrow \chi$ in $L^p(\Omega ;\mathbb {R}^2)$ for all $p < \infty$ observe that due to the $L^\infty$ bound on the sequence $(\chi _n)_n$ it is enough to show the convergence only in $L^1(\Omega ;\mathbb {R}^2)$.

We start by showing that $\overline{\chi }_n \rightarrow \chi$ in $L^1(\Omega ;\mathbb {R}^2)$. Let us recall that $\chi = \nabla \varphi$. By (7.21) we get that

$$\begin{aligned} \Vert \overline{\chi }_n - \chi \Vert _{L^1(\Omega )} \leqq \Vert \mathrm {D}^{\mathrm {d}}\varphi _n - \nabla \varphi ^{\varepsilon _n} \Vert _{L^1(\Omega )} + \Vert \nabla \varphi ^{\varepsilon _n} - \nabla \varphi \Vert _{L^1(\Omega )} \end{aligned}$$

for n large enough. Using the bounds (7.17) and (7.19) (together with its analogue for discrete partial derivatives in the direction of $e_2$) already proven in Proposition 7.3, we obtain for the first term

$$\begin{aligned} \Vert \mathrm {D}^{\mathrm {d}}\varphi _n - \nabla \varphi ^{\varepsilon _n} \Vert _{L^1(\Omega )} \leqq C \lambda _n\rightarrow 0, \quad \text {as } n \rightarrow \infty . \end{aligned}$$

Moreover, from (7.5) we deduce that

$$\begin{aligned} \nabla \varphi (x) = \int _{\mathbb {R}^2} \nabla _{x} \eta (z,x) \overline{\varphi }(x) + \eta (z,x) \nabla \overline{\varphi }(x) \, {\mathrm {d}{z}}, \quad \text {for } x \in \Omega , \end{aligned}$$

where $\nabla _{x} \eta (z,x)$ denotes the gradient of $\eta$ with respect to the second group of variables. Together with (7.6), this yields

$$\begin{aligned} \begin{aligned} \int _{\Omega } |\nabla \varphi ^{\varepsilon _n}(x) - \nabla \varphi (x)| \, {\mathrm {d}{x}}&\leqq \int _{\Omega } \int _{\mathbb {R}^2} |\nabla _{x} \eta (z,x)|\ |\overline{\varphi }(x+\varepsilon _nz) - \overline{\varphi }(x)| \, {\mathrm {d}{z}} \, {\mathrm {d}{x}} \\&\quad + \int _{\Omega } \int _{\mathbb {R}^2} |\eta (z,x)|\ |\nabla \overline{\varphi }(x+\varepsilon _nz) - \nabla \overline{\varphi }(x)| \, {\mathrm {d}{z}} \, {\mathrm {d}{x}} \\&\leqq \Vert \nabla \eta \Vert _{L^\infty } \int _{B_R} \Vert \overline{\varphi }(\, \cdot +\varepsilon _nz) - \overline{\varphi }\Vert _{L^1(\Omega )} \, {\mathrm {d}{z}} \\&\quad + \Vert \eta \Vert _{L^\infty } \int _{B_R} \Vert \nabla \overline{\varphi }(\, \cdot +\varepsilon _nz) - \nabla \overline{\varphi }\Vert _{L^1(\Omega )} \, {\mathrm {d}{z}} \rightarrow 0 \end{aligned} \end{aligned}$$

as $n\rightarrow \infty$, where $R>0$ is a radius (independent of n and x) such that $z \mapsto \eta (z,x)$ is supported in $B_R$ and we have used the continuity of translations of $L^1$ functions. This concludes the proof that $\overline{\chi }_n \rightarrow \chi$ in $L^1(\Omega ;\mathbb {R}^2)$.

Hence, it remains to show that $\Vert \chi _n - \overline{\chi }_n \Vert _{L^1(\Omega )} \rightarrow 0$. Similarly as in Remark 2.5 we have that $|\chi _n - \overline{\chi }_n| \leqq C \delta _n|\overline{\chi }_n|^3$. Thus, by (7.22) we even have that $\chi _n - \overline{\chi }_n \rightarrow 0$ in $L^\infty (\Omega )$. $\square$

It remains to show that $\limsup _n H_n(\chi _n, \Omega ) \leqq Y[\eta ](\chi )$. We will achieve this by comparing the energies $H_n(\chi _n, \Omega )$ to the discrete Aviles–Giga-like energies from Proposition 7.3. This is the only part of the proof in which we require the scaling assumption $\frac{\delta _n^{5/2}}{\lambda _n} \rightarrow 0$.

Proposition 7.6

Assume that $\frac{\delta _n^{5/2}}{\lambda _n} \rightarrow 0$. Then,

$$\begin{aligned} H_n(\chi _n, \Omega ) = \frac{1}{2} \int _\Omega \frac{1}{\varepsilon _n} W(\mathrm {D}^{\mathrm {d}}\varphi _n) + \varepsilon _n| \Delta _{\mathrm {s}}^{\mathrm {d}}\varphi _n |^2 \, {\mathrm {d}{x}} + o_n(1). \end{aligned}$$

Proof

Step 1. (Estimate of $|W(\chi _n) - W(\mathrm {D}^{\mathrm {d}}\varphi _n)|$.) We prove that

$$\begin{aligned} \frac{1}{\varepsilon _n} \int _\Omega |W(\chi _n) - W(\mathrm {D}^{\mathrm {d}}\varphi _n)| \, {\mathrm {d}{x}} \rightarrow 0 \, . \end{aligned}$$

First, by (7.21) we have that

$$\begin{aligned} \begin{aligned} |W(\chi _n) - W(\mathrm {D}^{\mathrm {d}}\varphi _n)|&= \big | 1 - |\chi _n|^2 + 1 - |\overline{\chi }_n|^2 \big | \big | |\overline{\chi }_n|^2 - |\chi _n|^2 \big | \\&\leqq \Big ( 2 \big | 1 - |\overline{\chi }_n|^2 \big | + \big | |\overline{\chi }_n|^2 - |\chi _n|^2 \big | \Big ) \big | |\overline{\chi }_n|^2 - |\chi _n|^2 \big | \, . \end{aligned} \end{aligned}$$

Next, as in Remark 2.5 we obtain that $|\chi _n - \overline{\chi }_n| \leqq C \delta _n|\overline{\chi }_n|^3$. In view of (7.22) and (2.13) we get

$$\begin{aligned} \begin{aligned}&\frac{1}{\sqrt{\varepsilon _n}} \big \Vert |\overline{\chi }_n|^2 - |\chi _n|^2 \big \Vert _{L^2(\Omega )}&= \frac{1}{\sqrt{\varepsilon _n}} \big \Vert | \overline{\chi }_n + \chi _n | | \overline{\chi }_n - \chi _n | \big \Vert _{L^2(\Omega )} \\&\quad \leqq C \frac{\delta _n}{\sqrt{\varepsilon _n}} = C \bigg ( \frac{\delta _n^{5/2}}{\lambda _n} \bigg )^{\! 1/2} \rightarrow 0. \end{aligned} \end{aligned}$$

Moreover, by (7.7) and Proposition 7.3,

$$\begin{aligned} \frac{1}{\varepsilon _n} \big \Vert 1 - |\overline{\chi }_n|^2 \big \Vert _{L^2(\Omega )}^2 = \frac{1}{\varepsilon _n} \int _\Omega W(\mathrm {D}^{\mathrm {d}}\varphi _n) \, {\mathrm {d}{x}} \leqq C. \end{aligned}$$

In conclusion, by Hölder’s inequality,

$$\begin{aligned} \frac{1}{\varepsilon _n} \int _\Omega |W(\chi _n) - W(\mathrm {D}^{\mathrm {d}}\varphi _n)| \, {\mathrm {d}{x}} \leqq \Big ( \tfrac{2}{\sqrt{\varepsilon _n}} \big \Vert 1 - |\overline{\chi }_n|^2 \big \Vert _{L^2(\Omega )} + o_n(1) \Big ) \cdot o_n(1) \rightarrow 0. \end{aligned}$$

Step 2. (Estimate of $|W^\mathrm {d}(\chi _n) - W(\chi _n)|$.) We prove that

$$\begin{aligned} \frac{1}{\varepsilon _n} \int _\Omega |W^\mathrm {d}(\chi _n) - W(\chi _n)| \, {\mathrm {d}{x}} \rightarrow 0. \end{aligned}$$

As in (2.36) we have that

$$\begin{aligned} \begin{aligned}&\big | \sqrt{W^\mathrm {d}}(\chi _n^{i,j}) - \sqrt{W}(\chi _n^{i,j}) \big |\\&\quad \leqq \frac{1}{2} \big | (\chi _{1,n}^{i,j} + \chi _{1,n}^{i-1,j}) \lambda _n\partial ^{\mathrm {d}}_1 \chi _{1,n}^{i-1,j} + (\chi _{2,n}^{i,j} + \chi _{2,n}^{i,j-1}) \lambda _n\partial ^{\mathrm {d}}_2 \chi _{2,n}^{i,j-1} \big | \\&\quad \leqq C \lambda _n\big ( |\mathrm {D}^{\mathrm {d}}\chi _n^{i-1,j}| + |\mathrm {D}^{\mathrm {d}}\chi _n^{i,j-1}| \big ) \, , \end{aligned} \end{aligned}$$

where we have used (7.22). By writing $\chi _n$ in terms of $\overline{\chi }_n$ and using the 1-Lipschitz continuity of the map $s \mapsto \frac{2}{\sqrt{\delta _n}} \sin (\frac{\sqrt{\delta _n}}{2} s)$ we get that $|\mathrm {D}^{\mathrm {d}}\chi _n| \leqq |\mathrm {D}^{\mathrm {d}}\overline{\chi }_n| = |\mathrm {D}^{\mathrm {d}}\mathrm {D}^{\mathrm {d}}\varphi _n|$. Let us observe that by the bounds (7.11) and (7.18) we have that $\Vert \mathrm {D}^{\mathrm {d}}\mathrm {D}^{\mathrm {d}}\varphi _n \Vert _{L^2(\mathbb {R}^2)} \leqq \frac{C}{\sqrt{\varepsilon _n}}$ and, as a consequence, by (2.13),

$$\begin{aligned} \frac{1}{\sqrt{\varepsilon _n}} \big \Vert \sqrt{W^\mathrm {d}}(\chi _n) - \sqrt{W}(\chi _n) \big \Vert _{L^2(\Omega )} \leqq C \frac{\lambda _n}{\varepsilon _n} \rightarrow 0. \end{aligned}$$

Writing $W^\mathrm {d}- W = \big ( 2 \sqrt{W} + (\sqrt{W^\mathrm {d}} - \sqrt{W}) \big ) \big (\sqrt{W^\mathrm {d}} - \sqrt{W} \big )$ we infer that

$$\begin{aligned} \frac{1}{\varepsilon _n} \int _\Omega |W^\mathrm {d}(\chi _n) - W(\chi _n)| \, {\mathrm {d}{x}} \leqq \Big ( \tfrac{2}{\sqrt{\varepsilon _n}} \big \Vert \sqrt{W(\chi _n)} \Vert _{L^2(\Omega )} + o_n(1) \Big ) \cdot o_n(1) \rightarrow 0, \end{aligned}$$

where we have used that that $\frac{1}{\sqrt{\varepsilon _n}} \Vert \sqrt{W(\chi _n)} \Vert _{L^2(\Omega )} \leqq C$ by (7.7), Proposition 7.3, and Step 7.

Step 3. (Estimate of $\big | |A^\mathrm {d}(\chi _n)|^2 - |\Delta _{\mathrm {s}}^{\mathrm {d}}\varphi _n |^2 \big |$) We prove that

$$\begin{aligned} \varepsilon _n\int _{\Omega } \big | |A^\mathrm {d}(\chi _n)|^2 - |\Delta _{\mathrm {s}}^{\mathrm {d}}\varphi _n |^2 \big | \, {\mathrm {d}{x}} \rightarrow 0. \end{aligned}$$

To show this we observe that

$$\begin{aligned} \big | |A^\mathrm {d}(\chi _n)|^2 - |\Delta _{\mathrm {s}}^{\mathrm {d}}\varphi _n |^2 \big | = \big | A^\mathrm {d}(\chi _n) + \Delta _{\mathrm {s}}^{\mathrm {d}}\varphi _n \big | \big | A^\mathrm {d}(\chi _n) - \Delta _{\mathrm {s}}^{\mathrm {d}}\varphi _n |, \end{aligned}$$

where, by (7.21),

$$\begin{aligned} \big | A^\mathrm {d}(\chi _n)^{i,j} + \Delta _{\mathrm {s}}^{\mathrm {d}}\varphi _n^{i,j} \big |&\leqq |\mathrm {D}^{\mathrm {d}}{\widetilde{\chi }}_n^{i-1,j}| + |\mathrm {D}^{\mathrm {d}}{\widetilde{\chi }}_n^{i,j-1}| \nonumber \\&\quad + |\mathrm {D}^{\mathrm {d}}\overline{\chi }_n^{i-1,j}| + |\mathrm {D}^{\mathrm {d}}\overline{\chi }_n^{i,j-1}| \end{aligned}$$

(7.23)

and

$$\begin{aligned} \big | A^\mathrm {d}(\chi _n) - \Delta _{\mathrm {s}}^{\mathrm {d}}\varphi _n |&\leqq |\mathrm {D}^{\mathrm {d}}{\widetilde{\chi }}_n^{i-1,j} - \mathrm {D}^{\mathrm {d}}\overline{\chi }_n^{i-1,j}| \nonumber \\&\quad + |\mathrm {D}^{\mathrm {d}}{\widetilde{\chi }}_n^{i,j-1} - \mathrm {D}^{\mathrm {d}}\overline{\chi }_n^{i,j-1}|. \end{aligned}$$

(7.24)

To estimate the right-hand side in (7.23) we use the 1-Lipschitz continuity of the map $s \mapsto \frac{1}{\sqrt{\delta _n}} \sin (\sqrt{\delta _n} s)$ to obtain that $|\mathrm {D}^{\mathrm {d}}{\widetilde{\chi }}_n| \leqq |\mathrm {D}^{\mathrm {d}}\overline{\chi }_n| = | \mathrm {D}^{\mathrm {d}}\mathrm {D}^{\mathrm {d}}\varphi _n|$. As in Step 7 we have that $\Vert \mathrm {D}^{\mathrm {d}}\mathrm {D}^{\mathrm {d}}\varphi _n \Vert _{L^2(\mathbb {R}^2)} \leqq \frac{C}{\sqrt{\varepsilon _n}}$ by the bounds (7.11) and (7.18) in the proof of Proposition 7.3. As a consequence,

$$\begin{aligned} \big \Vert A^\mathrm {d}(\chi _n) + \Delta _{\mathrm {s}}^{\mathrm {d}}\varphi _n \big \Vert _{L^2(\Omega )} \leqq \frac{C}{\sqrt{\varepsilon _n}}. \end{aligned}$$

To estimate the right-hand side in (7.24) we proceed similarly as in Step 6 in Section 6. Specifically, as in (6.20) we have that

$$\begin{aligned} | \partial ^{\mathrm {d}}_k {\widetilde{\chi }}_{h,n} - \partial ^{\mathrm {d}}_k \overline{\chi }_{h,n}| \leqq C \delta _n\big ( |\overline{\chi }_{h,n}|^2 + |\overline{\chi }_{h,n}^{{\varvec{\cdot }}+ e_k}|^2 \big ) |\partial ^{\mathrm {d}}_k \overline{\chi }_{h,n}| \leqq C \delta _n|\partial ^{\mathrm {d}}_{kh} \varphi _n| \end{aligned}$$

for $k,h = 1,2$, where the last inequality is due to (7.21) and (7.22). Using again that $\Vert \mathrm {D}^{\mathrm {d}}\mathrm {D}^{\mathrm {d}}\varphi _n \Vert _{L^2(\mathbb {R}^2)} \leqq \frac{C}{\sqrt{\varepsilon _n}}$, we obtain that $\Vert \partial ^{\mathrm {d}}_k {\widetilde{\chi }}_{h,n} - \partial ^{\mathrm {d}}_k \overline{\chi }_{h,n} \Vert _{L^2(\Omega )} \leqq C \frac{\delta _n}{\sqrt{\varepsilon _n}}$. This yields

$$\begin{aligned} \Vert A^\mathrm {d}(\chi _n) - \Delta _{\mathrm {s}}^{\mathrm {d}}\varphi _n \Vert _{L^2(\Omega )} \leqq C \frac{\delta _n}{\sqrt{\varepsilon _n}}. \end{aligned}$$

Finally, our estimates lead to

$$\begin{aligned} \varepsilon _n\int _{\Omega } \big | |A^\mathrm {d}(\chi _n)|^2 - |\Delta _{\mathrm {s}}^{\mathrm {d}}\varphi _n |^2 \big | \, {\mathrm {d}{x}} \leqq C \delta _n\rightarrow 0. \end{aligned}$$

Recalling that

$$\begin{aligned} H_n(\chi _n, \Omega ) = \frac{1}{2} \int _\Omega \frac{1}{\varepsilon _n} W^\mathrm {d}(\chi _n) + \varepsilon _n|A^\mathrm {d}(\chi _n)|^2 \, {\mathrm {d}{x}}, \end{aligned}$$

Steps 7–7 yield the claim of the proposition. $\square$

Thanks to (7.7) and Propositions 7.3 and 7.6, we have proved (7.9). Since by Proposition 7.5 the sequence $(\chi _n)_n$ moreover satisfies (7.2), this concludes the proof of Theorem 4.1-iii).