Semicartesian surfaces and the relaxed area of maps from the plane to the plane with a line discontinuity

Abstract

In this paper, we estimate the area of the graph of a map ${\mathbf{u}}: \varOmega \subset \mathbb {R}^2\rightarrow \mathbb {R}^2$ discontinuous on a segment $J_{\mathbf{u}}$, with $J_{\mathbf{u}}$ either compactly contained in the bounded open set $\Omega$, or starting and ending on $\partial \Omega$. We characterize $\overline{{\mathcal {A}}}^\infty ({\mathbf{u}},\varOmega )$, the relaxed area functional in a sort of uniform convergence, in terms of the infimum of the area of those surfaces in $\mathbb {R}^3$ spanning the graphs of the traces of ${\mathbf{u}}$ on the two sides of $J_{\mathbf{u}}$ and having what we have called a semicartesian structure. We exhibit examples showing that $\overline{{\mathcal {A}}}({\mathbf{u}},\varOmega )$, the relaxed area in $L^1(\varOmega ; \mathbb {R}^2)$, may depend on the values of ${\mathbf{u}}$ far from $J_{\mathbf{u}}$ and also on the relative position of $J_{\mathbf{u}}$ with respect to $\partial \varOmega$. These examples confirm the highly non-local behavior of $\overline{{\mathcal {A}}}({\mathbf{u}},\cdot )$ and justify the interest in the study of $\overline{{\mathcal {A}}}^\infty$. Finally we prove that $\overline{{\mathcal {A}}}({\mathbf{u}},\cdot )$ is not subadditive for a rather large class of discontinuous maps ${\mathbf{u}}$.

1 Introduction

Given a bounded open set $\varOmega \subset \mathbb {R}^2 = \mathbb {R}^2_{(x,y)}$ and a map ${\mathbf{v}}:=(v_1,v_2): \varOmega \rightarrow \mathbb {R}^2 = \mathbb {R}^2_{(\xi ,\eta )}$ of class ${\mathcal {C}}^1$, the (nonparametric) area functional is defined as

$$\begin{aligned} {\mathcal {A}}({\mathbf{v}},\varOmega ):= \int _{\varOmega } |{\mathcal {M}}(\nabla {\mathbf{v}})|\,\hbox {d}x\,\hbox {d}y, \end{aligned}$$

where $\nabla {\mathbf{v}}$ is the Jacobian matrix of ${\mathbf{v}}$, ${\mathcal {M}}(\zeta )$ is the vector of $\mathbb {R}^6$ having as entries the determinant of all minors¹ of the $(2\times 2)$-matrix $\zeta$, and $|\cdot |$ denotes the Euclidean norm: hence, $\vert {\mathcal {M}}(\nabla {\mathbf{v}})\vert = \sqrt{1 + \vert \nabla v_1 \vert ^2 + \vert \nabla v_2 \vert ^2 + \left( \partial _x v_1 \partial _y v_2 - \partial _y v_1 \partial _x v_2\right) ^2}$.

The functional ${\mathcal {A}}(\cdot ,\Omega )$ is polyconvex [5] and ${\mathcal {A}}({\mathbf{v}},\Omega )$ is the area of

$$\begin{aligned} \mathrm{graph}({\mathbf{v}}):=\{(x,y,\xi ,\eta ): (x,y)\in \varOmega ,\, (\xi ,\eta )={\mathbf{v}}(x,y)\} \subset \mathbb {R}^4:=\mathbb {R}^2_{(x,y)}\times \mathbb {R}^2_{(\xi ,\eta )}, \end{aligned}$$

a smooth two-dimensional manifold of codimension two.

For the purposes of the modern calculus of variations, it is useful to extend the area functional also to non-smooth maps. A rather natural idea consists in considering its $L^1$-lower semicontinuous envelope (or $L^1$-relaxed functional) [1, 7, 9, 10]; thus, for any ${\mathbf{v}}\in L^1(\varOmega ;\mathbb {R}^2)$, we set

$$\begin{aligned} \overline{{\mathcal {A}}}({\mathbf{v}},\Omega ):= \inf \left\{ \liminf _{h\rightarrow +\infty } {\mathcal {A}}({\mathbf{v}}_h,\varOmega )\right\} , \end{aligned}$$

where the infimum is taken among all sequences $({\mathbf{v}}_h)\subset {\mathcal {C}}^1(\varOmega ;\mathbb {R}^2)$ converging to ${\mathbf{v}}$ in $L^1(\varOmega ;\mathbb {R}^2)$. The aim of the present paper is to study $\overline{{\mathcal {A}}}({\mathbf{v}},\Omega )$ for certain classes of non-smooth maps ${\mathbf{v}}$. As we shall see, another (i.e., with respect to a different and stronger convergence) relaxed functional will be of interest, in this two-codimensional situation.

In [1, Theorem 3.7] it is proven that the domain of $\overline{{\mathcal {A}}}(\cdot ,\Omega )$ is contained² in the space $\mathrm{BV}(\varOmega ;\mathbb {R}^2)$ of maps with bounded variation in $\Omega$, and for any ${\mathbf{v}}\in \mathrm{BV}(\varOmega ;\mathbb {R}^2)$ it turns out that

where $\nabla {\mathbf{v}}$ and $D^s {\mathbf{v}}$ denote the absolutely continuous and the singular part of the distributional gradient $D {\mathbf{v}}$, respectively. Moreover, in [1, Theorem 6.4] the subset of $\mathrm{BV}(\varOmega ;\mathbb {R}^2)$ of all maps ${\mathbf{v}}$ for which

is characterized; from now on we shall use the symbol ${\mathcal {A}}$ in place of $\overline{{\mathcal {A}}}$ to denote the area of the graph of a map in this class. It is worth to notice that $\overline{{\mathcal {A}}}({\mathbf{v}},\varOmega )=\int _{\varOmega }|{\mathcal {M}}(\nabla {\mathbf{v}})|\, \text{ d }x\, \text{ d }y<+\infty$ for every ${\mathbf{v}}\in H^1(\varOmega ;\mathbb {R}^2)$.

One of the major issues on the functional $\overline{{\mathcal {A}}}({\mathbf{v}}, \cdot )$ is its non-subadditivity [7]. In [1, Theorems 4.1 and 5.1] the authors exhibit two examples of maps ${\mathbf{v}}$ for which there exist three bounded open sets $\varOmega _1$, $\varOmega _2$, and $\varOmega _3$ such that $\varOmega _3 \subset \subset \varOmega _1 \cup \varOmega _2$ and

$$\begin{aligned} \overline{{\mathcal {A}}}({\mathbf{v}},\varOmega _3)> \overline{{\mathcal {A}}}({\mathbf{v}},\varOmega _1)+\overline{{\mathcal {A}}}({\mathbf{v}},\varOmega _2). \end{aligned}$$

(1.1)

In the first theorem ${\mathbf{v}}={\mathbf{u}}_{T}\in \mathrm{BV}(\varOmega ;\mathbb {R}^2)$, a piecewise constant map taking three non-collinear values around a triple point, while in the second one³ ${\mathbf{v}}(x,y)= {{\mathbf{u}}_V}(x,y):=\frac{(x,y)}{|(x,y)|}$ (vortex map), and thus ${\mathbf{v}}\in W^{1,p}(\varOmega ;\mathbb {R}^2)$ for any $p \in [1, 2)$. Notice that inequality (1.1) implies that $\overline{{\mathcal {A}}}({\mathbf{v}},\Omega )$ cannot be written as an integral, over $\Omega$, of a local integrand, integrated with respect to some measure; recall that, on the contrary, this integral representation holds in codimension one (see [6, 11, 12]).

In [3] the authors provide an upper bound for $\overline{{\mathcal {A}}}({\mathbf{u}}_{T}, \varOmega )$ that improves the estimate of [1]. They are able to control the singular contribution of the relaxed area functional, namely

$$\begin{aligned} \overline{{\mathcal {A}}}({\mathbf{u}}_T, \varOmega ) - \int _{\varOmega } |{\mathcal {M}}(\nabla {\mathbf{u}}_T)|\,\hbox {d}x\, \hbox {d}y, \end{aligned}$$

through the area of a suitable graph-type area-minimizing two-dimensional surface of codimension one, entangled at the triple point with two other similar surfaces.

In [4], the authors explore further the idea of estimating the above-mentioned singular contribution through the area of solutions of a suitable Plateau’s-type problem in $\mathbb {R}^3$. More specifically, they study the case of a map ${\mathbf{u}}$ that is regular enough out of a simple smooth jump curve $J_{\mathbf{u}}$ compactly contained in $\varOmega$. Then they consider the closed curve $\varGamma = \varGamma [{\mathbf{u}}]\subset \mathbb {R}^3$, supposed to be simple, obtained as the union of the graphs of the traces of ${\mathbf{u}}$ on the two sides of $J_{\mathbf{u}}$; the regularity of ${\mathbf{u}}$ implies the existence of an area-minimizing immersion $X_{\min }\in {\mathcal {C}}^2(\overline{B};\mathbb {R}^3)\cap {\mathcal {C}}^\omega (B;\mathbb {R}^3)$, mapping the boundary of the unit disk B monotonically onto $\varGamma$, see for example [13] and [8]. In [4, Theorem 4.1] it is proven that, if $\Sigma _{\min }:=X_{\min }(B)$ admits a semicartesian parametrization, then

$$\begin{aligned} \overline{{\mathcal {A}}}({\mathbf{u}}, \varOmega ) \le \int _{\varOmega } |{\mathcal {M}}(\nabla {\mathbf{u}})|\,\text{ d }x\, \text{ d }y + {\mathcal {H}}^2(\Sigma _{\min }), \end{aligned}$$

(1.2)

where ${\mathcal {H}}^2$ denotes the two-dimensional Hausdorff measure. A map $\varPhi : O\rightarrow \mathbb {R}^3$, $O\subset \mathbb {R}^2_{(t,s)}$ a bounded, open, connected and simply-connected set, is said to be semicartesian if it is the identity in the first coordinate, that is if $\varPhi (t,s):=(t, \varPhi _2(t,s), \varPhi _3(t,s))$, see Definition 2.3 for more details. In [4, Theorem 5.1] sufficient conditions on $\varGamma$, i.e., $\varGamma$ analytic with further non-degeneracy hypotheses, are given in order that $\Sigma _{\min }$ admits a semicartesian parametrization, and it is also conjectured that (1.2) could be an equality, at least when $J_{\mathbf{u}}$ is far enough from $\partial \varOmega$. With the methods employed in [4], it seems not easy to weaken the analyticity and non-degeneracy assumptions (a part from the case when $\varGamma$ admits a graph-type solution of the corresponding Plateau’s problem).

In this paper, we continue the analysis on the singular contribution of the nonparametric area functional for maps ${\mathbf{u}}$ having a line discontinuity $J_{\mathbf{u}}$, in terms of suitable area-minimizing semicartesian surfaces. We shall analyze both the case when $J_{\mathbf{u}}$ is compactly contained in $\varOmega$ as well as when both its endpoints belong to $\partial \varOmega$. These two cases are quite different from each other; in particular, as we shall see, the latter turns out to be related to minimal surfaces with a partially free boundary. Notice that we do not suppose a priori that the union of the graphs of the traces of ${\mathbf{u}}$ on $J_{\mathbf{u}}$ is a Jordan curve. Since we deal with maps with Lipschitz traces, even when $\varGamma$ is a Jordan curve, it does not satisfy the sufficient conditions of [4] that guarantee the existence of a semicartesian parametrization for a solution of the corresponding Plateau’s problem.

Before stating our main results, we need to fix some notation and give some definitions, referring to Sect. 2 for the details.

Given two maps $\gamma ^\pm \in \mathrm{Lip}([a,b];\mathbb {R}^2)$, we consider their graphs $\varGamma ^\pm \subset \mathbb {R}^3:=\mathbb {R}_t\times \mathbb {R}^2_{(\xi ,\eta )}$. Let $\mathrm{R}:=(a,b)\times (-1,1)\subset \mathrm{R}^2_{(t,s)}$. We denote by $\mathrm{semicart}(\mathrm{R};\varGamma ^-,\varGamma ^+)$ the class of semicartesian maps on $\mathrm{R}$ spanning $\varGamma :=\varGamma ^-\cup \varGamma ^+$, that is the class of maps $\varPhi \in H^1(\mathrm{R};\mathbb {R}^3)$ such that

$$\begin{aligned} \varPhi (t,s):=(t,\varPhi _2(t,s),\varPhi _3(t,s)), \qquad \varPhi (t,\pm 1)=(t, \gamma ^\pm (t)). \end{aligned}$$

In particular, $\varPhi (\mathrm{R})$ is a surface that intersects any plane $\{t\}\times \mathbb {R}^2_{(\xi ,\eta )}$ in a (not necessarily simple) curve connecting the points $(t, \gamma ^-(t))$ and $(t, \gamma ^+(t))$, for any $t\in [a,b]$.

Since $\mathrm{semicart}(\mathrm{R}; \varGamma ^-,\varGamma ^+)$ is non-empty (Lemma 2.8), we can define

$$\begin{aligned} m(\mathrm{R};\varGamma ^-,\varGamma ^+):=\inf _{\varPhi \in \mathrm{semicart}(\mathrm{R}; \varGamma ^-,\varGamma ^+)}\int _\mathrm{R}|\partial _t\varPhi \wedge \partial _s\varPhi |\,\hbox {d}t\,\hbox {d}s. \end{aligned}$$

(1.3)

If $\varGamma$ is a closed (not necessarily simple) curve, we can consider also another class of maps. Let

$$\begin{aligned} D:=\{(t,s)\in \mathbb {R}^2: t\in (a,b),\,s\in (\sigma ^-(t),\sigma ^+(t))\}, \end{aligned}$$

with $\sigma ^\pm \in \mathrm{Lip}([a,b])$, $\sigma ^-(t)<0$ and $\sigma ^+(t)>0$ for $t\in (a,b)$ and $\sigma ^\pm (a)=0=\sigma ^\pm (b)$. Then $\mathrm{semicart}(D;\varGamma ^-,\varGamma ^+)$ denotes the class of semicartesian maps defined on D and spanning $\varGamma :=\varGamma ^-\cup \varGamma ^+$, that is maps $\varPhi \in H^1(D;\mathbb {R}^3)$ such that

$$\begin{aligned} \varPhi (t,s):=(t,\varPhi _2(t,s),\varPhi _3(t,s)), \qquad \varPhi (t,\sigma ^\pm (t))=(t,\gamma ^\pm (t)). \end{aligned}$$

For such a $\varPhi$, the image $\varPhi (D)$ is a surface whose intersection with any plane $\{t\}\times \mathbb {R}^2_{(\xi ,\eta )}$, t belonging to the open interval (a, b), is a curve connecting $(t, \gamma ^-(t))$ and $(t, \gamma ^+(t))$, but whose intersection with the plane $\{a\}\times \mathbb {R}^2_{(\xi ,\eta )}$ (resp. with $\{b\}\times \mathbb {R}^2_{(\xi ,\eta )}$) is the singleton $(a,\gamma ^-(a))$ (resp. $(b,\gamma ^-(b))$).

Also $\mathrm{semicart}(D;\varGamma ^-,\varGamma ^+)$ is non-empty and then we can define

$$\begin{aligned} m(D;\varGamma ^-,\varGamma ^+):=\inf _{\varPhi \in \mathrm{semicart}(D; \varGamma ^-,\varGamma ^+)}\int _D |\partial _t\varPhi \wedge \partial _s\varPhi |\,\hbox {d}t\,\hbox {d}s. \end{aligned}$$

We observe that when $\varGamma$ is closed, $m(\mathrm{R};\varGamma ^-,\varGamma ^+)\le m(D;\varGamma ^-,\varGamma ^+)$, since a surface that is image of a map in $\mathrm{semicart}(D; \varGamma ^-, \varGamma ^+)$ can be obtained also as the image of a map in $\mathrm{semicart}(\mathrm{R}; \varGamma ^-, \varGamma ^+)$ (see (2.6)). If $\varGamma$ is closed and simple we could ask about the relations between $m(D;\varGamma ^-,\varGamma ^+)$, $m(\mathrm{R};\varGamma ^-,\varGamma ^+)$, and $a(\varGamma )$, the area of a solution of the classical Plateau’s problem for $\varGamma$. In general $a(\varGamma )\le m(D;\varGamma ^-,\varGamma ^+)$, but one could expect also that the equal sign holds (see Remark 2.11), while we exhibit in Example 2.13 a curve $\varGamma$ for which

$$\begin{aligned} m(\mathrm{R};\varGamma ^-,\varGamma ^+)<a(\varGamma ). \end{aligned}$$

In the study of the $L^1$-lower semicontinuous envelope of the area functional ${\mathcal {A}}(\cdot ,\Omega )$, it would be important to have an $L^1$-lower semicontinuity result for $m(\mathrm{R}; \cdot , \cdot )$, compare also with Remark 4.7. More precisely, if $\varGamma ^\pm :=\mathrm{graph} (\gamma ^\pm )$, $\varGamma ^\pm _h:=\mathrm{graph} (\gamma ^\pm _h)$ with $\gamma ^\pm , \gamma ^\pm _h \in \mathrm{Lip}([a,b];\mathbb {R}^2)$, it would be desirable to prove that

$$\begin{aligned} m(\mathrm{R};\varGamma ^-,\varGamma ^+)\le \liminf _{h\rightarrow +\infty } m(\mathrm{R};\varGamma ^-_h,\varGamma ^+_h) \end{aligned}$$

(1.4)

whenever $\gamma ^\pm _h \rightarrow \gamma ^\pm$ in $L^1((a,b);\mathbb {R}^2)$. We are able to prove (1.4) only under the further assumption that

$$\begin{aligned} \sup _{h \in \mathbb N} ||\dot{\gamma }^\pm _h||_{L^\infty ((a,b);\mathbb {R}^2)} < +\infty . \end{aligned}$$

(1.5)

This is, in some sense, coherent with the lower semicontinuity of $a(\cdot )$ with respect to the Fréchet convergence (see [8, 13, §301]); we shall show that $a(\cdot )$ is not lower semicontinuous with respect to the $L^1$-convergence, see Example 4.8 for the details. Proving the validity of (1.4) without assuming (1.5) seems not to be easy and would imply a characterization of $\overline{{\mathcal {A}}}({\mathbf{u}},\Omega )$ for certain non-smooth maps ${\mathbf{u}}$. The lack of a proof of (1.4) under the mere $L^1$-convergence forced us to define another extension of the functional ${\mathcal {A}}(\cdot ,\Omega )$ with respect to a stronger notion of convergence, that we now describe.

Definition 1.1

(Uniform convergence out of a closed set) Let ${\mathbf{v}}\in \mathrm{BV}(\Omega ; \mathbb {R}^2)$ and $J\subset \varOmega$ be a closed set with zero Lebesgue measure. A sequence $({\mathbf{v}}_h)\subset L^1(\varOmega ;\mathbb {R}^2)$ is said to converge to ${\mathbf{v}}$ uniformly out of J, if ${\mathbf{v}}_h \rightarrow {\mathbf{v}}$ uniformly in any compact set of $\varOmega {\setminus } J$, as $h \rightarrow +\infty$.

We shall always consider maps ${\mathbf{u}}\in \mathrm{BV}(\Omega ; \mathbb {R}^2)$ so that, at any point of the (approximate) jump set [2], the approximate two-sided limits, denoted by ${\mathbf{u}}^\pm$, coincide with the pointwise two-sided limits, and from now on, with a small abuse of notation, $J_{\mathbf{u}}$ stands for the closure in $\Omega$ of the set $\{(x,y)\in \varOmega : {\mathbf{u}}^-(x,y) \ne {\mathbf{u}}^+(x,y)\}$.

We are now in a position to define another notion of relaxation of the area functional.

Definition 1.2

(The functional $\overline{{\mathcal {A}}}^\infty$) For any ${\mathbf{v}}\in \mathrm{BV}(\varOmega ;\mathbb {R}^2)$ we define

$$\begin{aligned} \overline{{\mathcal {A}}}^\infty ({\mathbf{v}}, \varOmega ):=\inf \left\{ \liminf _{h \rightarrow +\infty }{\mathcal {A}}({\mathbf{v}}_h,\varOmega )\right\} , \end{aligned}$$

where the infimum is taken among all sequences $({\mathbf{v}}_h)\subset {\mathcal {C}}^1(\varOmega ;\mathbb {R}^2)$ converging to ${\mathbf{v}}$ in $L^1(\varOmega ; \mathbb {R}^2)$ and uniformly out of $J_{{\mathbf{v}}}$.

It is clear that

$$\begin{aligned} \overline{{\mathcal {A}}}^\infty ({\mathbf{v}},\varOmega )\ge \overline{{\mathcal {A}}}({\mathbf{v}},\varOmega ), \qquad {\mathbf{v}}\in \mathrm{BV}(\varOmega ;\mathbb {R}^2). \end{aligned}$$

For every ${\mathbf{v}}$ in the domain of the functional $\overline{{\mathcal {A}}}(\cdot , \varOmega )$ it is also worth to define the singular parts

$$\begin{aligned} \begin{aligned} \overline{{\mathcal {A}}}_s({\mathbf{v}}, \varOmega )&:= \overline{{\mathcal {A}}}({\mathbf{v}},\varOmega ) - \int _\varOmega |{\mathcal {M}}(\nabla {\mathbf{v}})|\,\hbox {d}x \, \hbox {d}y, \\ \overline{{\mathcal {A}}}^\infty _s({\mathbf{v}}, \varOmega )&:= \overline{{\mathcal {A}}}^\infty ({\mathbf{v}},\varOmega ) - \int _\varOmega |{\mathcal {M}}(\nabla {\mathbf{v}})|\,\hbox {d}x \, \hbox {d}y. \end{aligned} \end{aligned}$$

The aim of this paper is to study the functionals $\overline{{\mathcal {A}}}_s$ and $\overline{{\mathcal {A}}}^\infty _s$, and also to characterize $\overline{{\mathcal {A}}}^\infty _s({\mathbf{u}}, \varOmega )$, for ${\mathbf{u}}$ in a suitable class of maps.

1.1 Main results

We shall consider two rather different cases: $\varOmega$ and ${\mathbf{u}}$ satisfying either condition I or condition II (see Definitions 2.15 and 2.16, respectively). Condition I takes into account maps ${\mathbf{u}}\in W^{1,\infty }(\varOmega {\setminus } J_{\mathbf{u}};\mathbb {R}^2)$ having as jump set $J_{\mathbf{u}}$ a horizontal segment with both endpoints belonging to $\partial \varOmega$; namely, the fracture “traverses” the whole domain $\Omega$. Condition II deals with maps ${\mathbf{u}}\in W^{1,\infty }(\varOmega {\setminus } J_{\mathbf{u}};\mathbb {R}^2)$ with $J_{\mathbf{u}}\subset \subset \varOmega$. We denote by

$$\begin{aligned} \varGamma ^\pm [{\mathbf{u}}] \end{aligned}$$

the graphs of the traces of ${\mathbf{u}}$ on the two sides of $J_{\mathbf{u}}$ (Sect. 2.1).

Our first result characterizes the lower semicontinuous envelope of ${\mathcal {A}}$ in the sense of Definition 1.2.

Theorem 1.3

(I: characterization of $\overline{{\mathcal {A}}}^\infty$) Let $\varOmega$ and ${\mathbf{u}}$ satisfy condition I. Then

$$\begin{aligned} \overline{{\mathcal {A}}}^\infty _s({\mathbf{u}},\varOmega ) = m(\mathrm{R};\varGamma ^-[{\mathbf{u}}],\varGamma ^+[{\mathbf{u}}]). \end{aligned}$$

(1.6)

The inequality $\overline{{\mathcal {A}}}^\infty _s({\mathbf{u}}, \varOmega ) \le m(\mathrm{R};\varGamma ^-[{\mathbf{u}}],\varGamma ^+[{\mathbf{u}}])$ is obtained using the same strategy proposed in [4], and it is proven in Proposition 3.1. The proof of the converse inequality (lower bound), presented in Sect. 4, is more interesting. As already noticed, in order to prove this inequality we use a lower semicontinuity result for $m(\mathrm{R};\cdot ,\cdot )$ with respect to a convergence that is stronger than the one induced by the $L^1$-convergence, see Definition 1.1. Since we miss the proof of the $L^1((a,b); \mathbb {R}^2)$-lower semicontinuity of $m(\mathrm{R}; \cdot , \cdot )$, we are not able to conclude the reasonable conjecture that $\overline{{\mathcal {A}}}_s({\mathbf{u}},\varOmega ) = m(\mathrm{R};\varGamma ^-[{\mathbf{u}}],\varGamma ^+[{\mathbf{u}}])$, for $\varOmega$ and ${\mathbf{u}}$ satisfying condition I and such that $\varGamma ^-[{\mathbf{u}}]\cap \varGamma ^+[{\mathbf{u}}]={\emptyset }$.

When $\varOmega$ and ${\mathbf{u}}$ satisfy condition II the situation is less clear. The proof of the next result is given in Proposition 5.1 and Theorem 6.1.

Theorem 1.4

(II: characterization of $\overline{{\mathcal {A}}}^\infty$) Let $\varOmega$ and ${\mathbf{u}}$ satisfy condition II. Then

$$\begin{aligned} \overline{{\mathcal {A}}}^\infty _s({\mathbf{u}},\varOmega ) = m(D;\varGamma ^-[{\mathbf{u}}],\varGamma ^+[{\mathbf{u}}]). \end{aligned}$$

What is interesting is that it may happen that

$$\begin{aligned} \overline{{\mathcal {A}}}_s({\mathbf{u}},\varOmega )< m(D;\varGamma ^-[{\mathbf{u}}],\varGamma ^+[{\mathbf{u}}]), \end{aligned}$$

and thus there exist $\varOmega$ and ${\mathbf{u}}$ for which

$$\begin{aligned} \overline{{\mathcal {A}}}({\mathbf{u}}, \varOmega )< \overline{{\mathcal {A}}}^\infty ({\mathbf{u}}, \varOmega ). \end{aligned}$$

In Sect. 7 we collect some examples proving that sequences $({\mathbf{u}}_h)$ converging to ${\mathbf{u}}$ only in $L^1(\varOmega ;\mathbb {R}^2)$ can be more “convenient” than any other sequence converging to ${\mathbf{u}}$ also uniformly out of $J_{\mathbf{u}}$. More specifically, in Sect. 7.1 we adapt to our case the construction used in [1, Lemma 5.3] concerning the area of the graph of the vortex map ${{\mathbf{u}}_V}$. The singular contribution of the area that we obtain can be interpreted as the area of a semicartesian parametrization defined on a suitable rectangle and spanning the graphs of the traces of ${\mathbf{u}}$ on a suitable extension $J_\mathrm{ext}$ of $J_{\mathbf{u}}$ that reaches $\partial \varOmega$. Example 7.4 proves that this construction can possibly provide an upper bound lower than $m(D;\varGamma ^-[{\mathbf{u}}], \varGamma ^+[{\mathbf{u}}])$. This suggests that it could be convenient to “extend” $J_{\mathbf{u}}$ up to the boundary by what we have called a virtual jump. As observed in Remark 7.5, we can manipulate the result in Example 7.4 and show that, if $J_{\mathbf{u}}$ has two connected components, the virtual jump could join one connected component to the other, instead of joining $J_{\mathbf{u}}$ to $\partial \varOmega$. In Sect. 7.2 we exhibit an example where it is even more convenient to consider a virtual jump connecting an internal point of $J_{\mathbf{u}}$ to $\partial \varOmega$.

All these examples reveal that the singular contribution of the area functional depends not only on the values of ${\mathbf{u}}$ near the jump set, but also on the values of ${\mathbf{u}}$ far from the jump and on the position of the jump with respect to $\partial \varOmega$, confirming the deep non-local behavior of $\overline{{\mathcal {A}}}({\mathbf{u}},\cdot )$.

Our last result, Theorem 8.1, concerns the non-subadditivity of $\overline{{\mathcal {A}}}$ with respect of the open set.

Theorem 1.5

There exist $\varOmega$ and ${\mathbf{u}}$ satisfying condition I such that $\overline{{\mathcal {A}}}({\mathbf{u}},\cdot )$ is not subadditive.

The class of maps ${\mathbf{u}}$ for which Theorem 8.1 holds is rather large, and those maps can be written explicitly (see (8.1)). From Theorem 1.5, one deduces that the non-local character of $\overline{{\mathcal {A}}}({\mathbf{u}}, \cdot )$ is not necessarily due to the presence of a vortex or of a triple junction, but it is a much more general fact.

We underline that in order to prove Theorem 1.5 we do not use the results in Sects. 4–6, but only the upper bound in Proposition 3.1, and some (limited) results concerning cartesian currents (Proposition 8.6).

The plan of the paper is the following. In Sect. 2 we introduce the definitions and the main properties concerning the semicartesian setting, and we fix the hypotheses on the class of maps we consider in this work. In Sects. 3 and 4, we prove the upper and the lower bound that, coupled together, yield the proof of Theorem 1.3. Section 4 contains also the discussion on the semicontinuity of $m(\mathrm{R};\cdot ,\cdot )$. Sections 5 and 6 deal with the case where $\varOmega$ and ${\mathbf{u}}$ satisfy condition II and contain the proof of the upper and the lower bound needed to prove Theorem 1.4. In Sect. 7, we exhibit some examples of pairs $(\varOmega ,{\mathbf{u}})$ for which $\overline{{\mathcal {A}}}({\mathbf{u}}, \varOmega )< \overline{{\mathcal {A}}}^\infty ({\mathbf{u}}, \varOmega )$. Finally in Sect. 8 we prove the non-subadditivity of $\overline{{\mathcal {A}}}({\mathbf{u}}, \cdot )$ for a rather large class of maps.

2 Semicartesian structure

Let us start with some definitions. From now on we take $a,b \in \mathbb {R}= \mathbb {R}_t$, with $a < b$.

Definition 2.1

(Union of two graphs) Let $\varGamma \subset \mathbb {R}^3=\mathbb {R}_t \times \mathbb {R}^2_{(\xi ,\eta )}$; we say that $\varGamma$ is union of two graphs on [a, b] if $\varGamma =\varGamma ^-\cup \varGamma ^+$, where $\varGamma ^\pm := \mathrm{graph}(\gamma ^\pm )$ with $\gamma ^\pm \in {\mathcal {C}}([a,b];\mathbb {R}^2)\cap \mathrm{Lip}_\mathrm{loc}((a,b); \mathbb {R}^2)$. We say that $\varGamma$ is union of two Lipschitz graphs on [a, b] if furthermore $\gamma ^\pm \in \mathrm{Lip}([a,b];\mathbb {R}^2)$.

Remark 2.2

Depending on the values of $\gamma ^\pm$ at $t=a$ and $t=b$, $\varGamma$ could be either a closed curve, or an open curve, or the union of two open curves. Notice that we do not exclude that $\gamma ^-(t)=\gamma ^+(t)$ for some $t\in (a,b)$. We shall be mostly interested in the cases when either $\varGamma$ is closed, or when $\gamma ^-(a) \ne \gamma ^+(a)$ and $\gamma ^-(b) \ne \gamma ^+(b)$. The latter case will be related to a partially free boundary problem.

Definition 2.3

(Semicartesian map) A semicartesian map on $O$ is a continuous map $\varPhi : \overline{O}\rightarrow \mathbb {R}^3$ of the form

$$\begin{aligned} \varPhi (t,s)=(t,\phi (t,s)) = (t, \phi _1(t,s), \phi _2(t,s)), \qquad (t,s) \in O, \end{aligned}$$

(2.1)

where

$$\begin{aligned} O:=\{(t,s) \in \mathbb {R}^2:~ \sigma ^-(t)<s<\sigma ^+(t),\,\,t\in (a,b)\}, \end{aligned}$$

with $\sigma ^\pm \in \mathcal C([a,b])\cap \mathrm{Lip}_\mathrm{loc}((a,b))$ and $\sigma ^-< \sigma ^+$ in (a, b).

If we need to stress the dependence on the functions $\sigma ^\pm$, we shall use the notation $O=[[\sigma ^-,\sigma ^+]]$.

Definition 2.4

(Semicartesian parametrizations) Given $\varGamma =\varGamma ^-\cup \varGamma ^+$ union of two graphs on [a, b], $\varGamma ^\pm :=\mathrm{graph}(\gamma ^\pm )$, a semicartesian parametrization spanning $\varGamma$ is a pair $(O, \varPhi )$ where $O=[[\sigma ^-,\sigma ^+]]$ and $\varPhi$ is a semicartesian map on $O$ satisfying the boundary condition

$$\begin{aligned} \varPhi (t,\sigma ^\pm (t))=(t, \gamma ^\pm (t)), \quad t \in [a,b]. \end{aligned}$$

We notice that, if $\gamma ^-(a)\ne \gamma ^+(a)$, the domain $O=[[\sigma ^-, \sigma ^+]]$ of a semicartesian parametrization $(O,\varPhi )$ spanning $\varGamma =\varGamma ^- \cup \varGamma ^+$ has to satisfy $\sigma ^-(a)<\sigma ^+(a)$. Similarly, if $\gamma ^-(b)\ne \gamma ^+(b)$, necessarily $\sigma ^-(b)<\sigma ^+(b)$. On the other hand, if $\gamma ^-(a)=\gamma ^+(a)$, we can in principle choose either a domain $O$ such that $\sigma ^-(a)=\sigma ^+(a)$ or such that $\sigma ^-(a)<\sigma ^+(a)$. The trace on the plane $\{t=a\}$ of the image of $\varPhi$ is, in the first case, just the point $(a,\gamma ^-(a))$; while in the second case, it is a not necessarily simple, closed, curve. Similar considerations are valid at $t=b$.

In the following, we shall need more regularity on $\varPhi$, since we need the area of a semicartesian parametrization to be finite; in Sects. 3 and 5 we will need $\varPhi$ and its derivatives to be square integrable, in order to build maps ${\mathbf{u}}_h \in H^1(\varOmega ;\mathbb {R}^2)$. This is, in some sense, coherent also with the classical theory of Plateau’s problem [8], where an area-minimizing immersion of the disk is found by minimizing the Dirichlet functional.

Finally we will fix special domains $O$. In Lemma 2.14 we will show that this can be done without loss of generality.

Definition 2.5

(The domains $\mathrm{R}$ and D) We set

$$\begin{aligned} \mathrm{R}:=(a,b)\times (-1,1), \end{aligned}$$

namely $\mathrm{R}=[[\sigma ^-_\mathrm{R}, \sigma ^+_\mathrm{R}]]$, with $\sigma ^-_\mathrm{R}\equiv -1$ and $\sigma ^+_\mathrm{R}\equiv 1$.

We also fix two maps $\sigma ^\pm \in \mathrm{Lip}([a,b])$ so that $\sigma ^-<\sigma ^+$ on (a, b) and

• $\sigma ^-(a)=\sigma ^+(a)=0$ and $\sigma ^\pm (t)= \mathcal {O}(t-a)$, for $t\in (a, a+\delta )$, $\delta >0$ small enough;

• $\sigma ^-(b)=\sigma ^+(b)=0$ and $\sigma ^\pm (t)= \mathcal {O}(b-t)$, for $t\in (b-\delta , b)$, $\delta >0$ small enough,⁴

and we define

$$\begin{aligned} D:= [[\sigma ^-,\sigma ^+]]. \end{aligned}$$

Definition 2.6

(The classes semicart) Let $\varGamma =\varGamma ^-\cup \varGamma ^+$ be union of two Lipschitz graphs on [a, b]. We set

$$\begin{aligned} \begin{aligned} \mathrm{semicart}(\mathrm{R}; \varGamma ^-, \varGamma ^+):=\{ \varPhi \in H^1(\mathrm{R};\mathbb {R}^3):\\ (\mathrm{R}, \varPhi ) \text{ semicartesian } \text{ parametrization } \text{ spanning } \varGamma \}, \end{aligned}\\ \begin{aligned} \mathrm{semicart}(D; \varGamma ^-, \varGamma ^+):=\{ \varPhi \in H^1(D;\mathbb {R}^3): \\ (D, \varPhi ) \text{ semicartesian } \text{ parametrization } \text{ spanning } \varGamma \}. \end{aligned} \end{aligned}$$

Remark 2.7

(Area integrand for semicartesian maps) For a semicartesian map $\varPhi$ as in (2.1) belonging either to $\mathrm{semicart}(\mathrm{R}; \varGamma ^-,\varGamma ^+)$ or to $\mathrm{semicart}(D; \varGamma ^-,\varGamma ^+)$, we have

$$\begin{aligned} |\partial _t \varPhi \wedge \partial _s \varPhi |=\sqrt{|\partial _s\phi |^2 + (\partial _t\phi _1 \partial _s \phi _2 - \partial _t \phi _2 \partial _s \phi _1)^2}. \end{aligned}$$

(2.2)

The area of a semicartesian parametrization is therefore

$$\begin{aligned} \int |\partial _t \varPhi \wedge \partial _s \varPhi |~\hbox {d}t\hbox {d}s =\int \sqrt{|\partial _s\phi |^2 + (\partial _t\phi _1 \partial _s \phi _2 - \partial _t \phi _2 \partial _s \phi _1)^2}\, \hbox {d}t\,\hbox {d}s , \end{aligned}$$

where the domain of integration of the integrals is either $\mathrm{R}$ or D. If in particular $\phi _1(t,s)=s$, the right hand side of (2.2) reduces obviously to $\sqrt{1+|\partial _t \phi _2|^2 + |\partial _s \phi _2|^2}$, namely the integrand of the area functional in the one-codimensional cartesian case.

Notice that, if $\varGamma ^\pm =\mathrm{graph}(\gamma ^\pm )$ with either $\gamma ^-(a)\ne \gamma ^+(a)$ or $\gamma ^-(b)\ne \gamma ^+(b)$, then the class $\mathrm{semicart}(D;\varGamma ^-, \varGamma ^+)$ is empty.

Lemma 2.8

Let $\varGamma =\varGamma ^-\cup \varGamma ^+$ be union of two Lipschitz graphs on [a, b]. Then

$$\begin{aligned} \mathrm{semicart}(\mathrm{R};\varGamma ^-, \varGamma ^+)\ne \emptyset . \end{aligned}$$

If in addition $\varGamma$ is closed, then also $\mathrm{semicart}(D;\varGamma ^-, \varGamma ^+)\ne \emptyset$.

Proof

Write $\varGamma ^\pm =\mathrm{graph} (\gamma ^\pm )$. Let us define the following $\mathbb {R}^2$-valued Lipschitz continuous linear interpolating map:

$$\begin{aligned} \ell _\mathrm{R}(t,s):=\frac{1-s}{2}\gamma ^-(t) +\frac{1+s}{2}\gamma ^+(t), \qquad (t,s) \in \overline{\mathrm{R}}. \end{aligned}$$

Then $\ell _\mathrm{R}(t, \pm 1)=\gamma ^\pm (t)$; thus the map $\varPhi _{\ell _\mathrm{R}}(t,s):=(t, \ell _\mathrm{R}(t,s))$ belongs to $\mathrm{semicart}(\mathrm{R};\varGamma ^-, \varGamma ^+)$.

Now, suppose that $\gamma ^-(a)=\gamma ^+(a)$ and $\gamma ^-(b)=\gamma ^+(b)$ (i.e., $\varGamma$ is closed) and define

$$\begin{aligned} \ell _D(t,s):= {\left\{ \begin{array}{ll} \frac{\sigma ^+(t) - s}{\sigma ^+(t)-\sigma ^-(t)}\gamma ^-(t) + \frac{s - \sigma ^-(t)}{\sigma ^+(t)-\sigma ^-(t)}\gamma ^+(t) \,\, &{}(t,s)\in D, \\ \gamma ^\pm (t) &{} s=\sigma ^\pm (t), \, t\in [a,b]. \end{array}\right. } \end{aligned}$$

Thus for $(t,s)\in D$

$$\begin{aligned} \begin{aligned} \partial _t \ell (t,s)&= \frac{\sigma ^+(t) - s}{\sigma ^+(t) - \sigma ^-(t)} \dot{\gamma }^-(t) + \frac{s - \sigma ^-(t)}{\sigma ^+(t) - \sigma ^-(t)} \dot{\gamma }^+(t) \\&\quad + \frac{(\gamma ^+(t) - \gamma ^-(t))(\dot{\sigma }^+(t)\sigma ^-(t) - \sigma ^+(t) \dot{\sigma }^-(t))}{(\sigma ^+(t) - \sigma ^-(t))^2}\\&\quad - \frac{s(\gamma ^+(t)-\gamma ^-(t))(\dot{\sigma }^+(t) - \dot{\sigma }^-(t))}{(\sigma ^+(t) - \sigma ^-(t))^2}, \\ \partial _s\ell (t,s)&= \frac{\gamma ^+(t) - \gamma ^-(t)}{\sigma ^+(t)-\sigma ^-(t)}. \end{aligned} \end{aligned}$$

Since $\gamma ^\pm \in \mathrm{Lip}([a,b];\mathbb {R}^2)$, the properties on $\sigma ^\pm$ in Definition 2.5 ensure that

$$\begin{aligned} \left| \frac{\gamma ^+(t) - \gamma ^-(t)}{\sigma ^+(t)-\sigma ^-(t)}\right| \le C < + \infty \, \qquad t \in (a,a+\delta ), \end{aligned}$$

and similarly for $t\in (b-\delta , b)$. Noticing also that $|\dot{\sigma }^+\sigma ^- - \sigma ^+ \dot{\sigma }^-|\le C(\sigma ^+ - \sigma ^-)$ (for a possibly different positive constant C), and recalling that $\gamma ^\pm$ and $\sigma ^\pm$ are Lipschitz continuous, we get that $\partial _t {\ell _D}$ and $\partial _s {\ell _D}$ are bounded. It follows that the map

$$\begin{aligned} \varPhi _{\ell _D}(t,s):=(t, {\ell _D}(t,s)) \end{aligned}$$

(2.3)

belongs to $W^{1,\infty }(D; \mathbb {R}^3)$, and in particular to $\mathrm{semicart}(D;\varGamma ^-,\varGamma ^+)$. $\square$

Remark 2.9

If $\gamma ^+(t)=\gamma ^-(t)$ for some $t\in (a,b)$, $\varPhi _{\ell _\mathrm{R}}$ (resp. $\varPhi _{\ell _D}$) maps the segment $\{t\}\times [-1,1]$ (resp. $\{t\}\times [\sigma ^-(t),\sigma ^+(t)]$) to the point $(t,\gamma ^+(t))$, hence it is not injective. More generally, a semicartesian map could be possibly not injective even if $\gamma ^-(t)\ne \gamma ^+(t)$ for every $t \in (a,b)$.

As a consequence of Lemma 2.8, we can introduce the following quantities.

Definition 2.10

(Minimal values m) Let $\varGamma =\varGamma ^-\cup \varGamma ^-$ be union of two Lipschitz graphs on [a, b]. We define

$$\begin{aligned} m(\mathrm{R};\varGamma ^-, \varGamma ^+):= \inf _{\varPhi \in \mathrm{semicart}(\mathrm{R};\varGamma ^-,\varGamma ^+)}\int _\mathrm{R}|\partial _t \varPhi \wedge \partial _s \varPhi |\,\hbox {d}t\,\hbox {d}s. \end{aligned}$$

(2.4)

If furthermore $\varGamma$ is closed we define

$$\begin{aligned} m(D;\varGamma ^-, \varGamma ^+):=\inf _{\varPhi \in \mathrm{semicart}(D;\varGamma ^-,\varGamma ^+)}\int _D |\partial _t \varPhi \wedge \partial _s \varPhi |\,\hbox {d}t\,\hbox {d}s. \end{aligned}$$

(2.5)

It is worthwhile to observe that (2.4) may become a partially free boundary problem, on the planes $\{t=a\}$ and $\{t=b\}$.

Trivially, if $\varGamma$ is closed, then

$$\begin{aligned} m(\mathrm{R};\varGamma ^-, \varGamma ^+)\le m(D;\varGamma ^-, \varGamma ^+). \end{aligned}$$

Indeed, supposing without loss of generality that $|\sigma ^\pm |< 1$, we can find, for any $\varPsi \in \mathrm{semicart}(D;\varGamma ^-, \varGamma ^+)$, a map $\varPhi \in \mathrm{semicart}(\mathrm{R};\varGamma ^-, \varGamma ^+)$ having the same area, defined as

$$\begin{aligned} \varPhi (t,s):= {\left\{ \begin{array}{ll} \varPsi (t,s) &{}\quad \mathrm{if~} (t,s) \in D,\\ \gamma ^+(t) &{}\quad \mathrm{if~} (t,s) \in \mathrm{R}{{\setminus }} D,\, s>0,\\ \gamma ^-(t) &{}\quad \mathrm{if~} (t,s) \in \mathrm{R}{{\setminus }} D,\, s<0. \end{array}\right. } \end{aligned}$$

(2.6)

Remark 2.11

(Semicartesian parametrizations and Plateau’s problem) The problem of the existence of a minimum in (2.4) and (2.5) seems to be open and requires further investigation. If $\varGamma = \varGamma ^-\cup \varGamma ^+$ is a closed simple curve, it is natural to compare $m(D;\varGamma ^-, \varGamma ^+)$ with the area $a(\varGamma )$ of a solution of the classical Plateau’s problem for $\varGamma$, that is an area-minimizing immersion of the disk, that maps the boundary of the disk onto $\varGamma$ monotonically, see for instance [13] and [8]. It is possible⁵ to see that

$$\begin{aligned} m(D;\varGamma ^-,\varGamma ^+)\ge a(\varGamma ). \end{aligned}$$

(2.7)

It is plausible that (2.7) holds with equal sign, and that $m(D;\varGamma ^-,\varGamma ^+)$ is actually a minimum. To substantiate these assertions, we recall that in [4] it is proven that solutions of the classical Plateau’s problem admit a semicartesian parametrization if $\varGamma$ is an analytic curve with further non-degeneracy properties at $(a,\gamma ^\pm (a))$ and $(b,\gamma ^\pm (b))$ (a case that does not fit in our setting). On the other hand, for what concerns the semicartesian maps defined on the rectangle $\mathrm{R}$, even assuming that $\varGamma$ is a closed simple curve, the existence of an area-minimizing semicartesian parametrization in $\mathrm{semicart}(\mathrm{R};\varGamma ^-,\varGamma ^+)$ does not follow from the existence of a solution for the Plateau’s problem. Indeed since the class of surfaces parametrized by maps in $\mathrm{semicart}(\mathrm{R};\varGamma ^-,\varGamma ^+)$ strictly contains (due to the free boundary on the planes $\{t=a\}$ and $\{t=b\}$) the ones parametrized by maps in $\mathrm{semicart}(D;\varGamma ^-,\varGamma ^+)$, we could expect that in general it contains also the class of surfaces considered in the classical setting. Moreover, we shall prove that possibly $a(\varGamma )> m(\mathrm{R};\varGamma ^-, \varGamma ^+)$: in Example 2.12 we build a semicartesian parametrization whose image is not in the class of surfaces considered for the classical Plateau’s problem; in Example 2.13 we exhibit $\gamma ^\pm \in \mathrm{Lip}([a,b];\mathbb {R}^2)$ such that the union of their graphs is a Jordan curve for which the semicartesian parametrization built in Example 2.12 has as area which is less than $a(\varGamma )$.

Fig. 1

Example 2.12. In the plane $\{0\}\times \mathbb {R}^2_{(\xi ,\eta )}$ we represent the curve $\mathrm{C}$; $\mathrm{C}$ is the projection of the curves $\varGamma ^\pm$ (in bold) on the plane $\{0\}\times \mathbb {R}^2_{(\xi ,\eta )}$. In light gray we draw the copies of $\mathrm{C}$ in the planes $\{t\}\times \mathbb {R}^2_{(\xi ,\eta )}$, $t\in [a,b]$. The surface $\varPhi (\mathrm{R})$ is the union of all portions of $\{t\}\times \mathrm{C}$ bounded by $(t, \gamma ^-(t))$ and $(t, \gamma ^+(t))$, when t varies in [a, b]

The next example is also strictly related to the construction made in Proposition 7.1.

Example 2.12

(Partially free boundary on $\{t=b\}$) Let $\gamma ^\pm \in \mathrm{Lip}([a,b];\mathbb {R}^2)$ and suppose that $\gamma ^-(a)=\gamma ^+(a)$; let us denote by $\mathrm{C}$ the (connected) set $\gamma ^-([a,b])\cup \gamma ^+([a,b])\subset \mathbb {R}^2_{(\xi ,\eta )}$. In Fig. 1 we draw a case when $\gamma ^+$ is not injective.

We want to define the map $\varPhi \in \mathrm{semicart}(\mathrm{R};\varGamma ^-,\varGamma ^+)$ which, for every $t\in (a,b)$, maps the segment $\{t\}\times [-1,1] \subset \mathrm{R}$ onto the portion of $\{t\}\times \mathrm{C}$ bounded by the points $(t,\gamma ^-(t))$ and $(t,\gamma ^+(t))$ and containing $(t,\gamma ^-(a))$.

If for convenience we parametrize $\mathrm{C}$ by a curve $\gamma \in \mathrm{Lip}([-1,1];\mathbb {R}^2)$, defined by

$$\begin{aligned} \gamma (\lambda ):= {\left\{ \begin{array}{ll} \gamma ^-(-(b-a)\lambda + a) &{}\quad \mathrm{if~} \lambda \in [-1,0], \\ \gamma ^+((b-a)\lambda + a) &{}\quad \mathrm{if~} \lambda \in (0,1], \end{array}\right. } \end{aligned}$$

so that $\gamma (-1)=\gamma ^-(b)$, $\gamma (0)=\gamma ^-(a)=\gamma ^+(a)$ and $\gamma (1)=\gamma ^+(b)$, then $\varPhi (\{t\}\times [-1,1])$ must be equal to $\left\{ (t, \gamma (\lambda )): \lambda \in \left[ -\frac{t-a}{b-a}, \frac{t-a}{b-a}\right] \right\}$. Thus we can define $\varPhi \in \mathrm{semicart}(\mathrm{R};\varGamma ^-,\varGamma ^+)$ as

$$\begin{aligned} \varPhi (t,s):=\left( t, \gamma \left( \frac{t-a}{b-a}s\right) \right) , \quad (t,s)\in \mathrm{R}. \end{aligned}$$

(2.8)

We observe that, if $\varGamma :=\mathrm{graph}(\gamma ^-) \cup \mathrm{graph}(\gamma ^+)$ is a closed simple curve, the surface $\varPhi (\mathrm{R})$ is not the image of an immersion of the disk mapping the boundary of the disk monotonically onto $\varGamma$, because $\varPhi (\partial R) = \varGamma \cup ( \{b\}\times {\mathrm{C}})$. Moreover,

$$\begin{aligned} m(\mathrm{R};\varGamma ^-, \varGamma ^+)\le \int _\mathrm{R}|\partial _t \varPhi \wedge \partial _s \varPhi |\, \text{ d }t \, \text{ d }s \le (b-a)\int _a^b \left( |\dot{\gamma }^-|+|\dot{\gamma }^+|\right) \, \hbox {d}t. \end{aligned}$$

Note that, if $\gamma ^\pm$ are injective, we have that $\varPhi (\mathrm{R})$ lies on the lateral part of the surface of the cylinder $(a,b)\times \mathrm{C}$.

We now exhibit maps $\gamma ^\pm \in \mathrm{Lip}([a,b];\mathbb {R}^2)$ so that $\varGamma :=\mathrm{graph}(\gamma ^-) \cup \mathrm{graph}(\gamma ^+)$ is a closed simple curve and $m(\mathrm{R};\varGamma ^-,\varGamma ^+)< a(\varGamma )$.

Example 2.13

($m(\mathrm{R};\varGamma ^-, \varGamma ^+)< a(\varGamma )$)Let $\rho$ be a positive real number with

$$\begin{aligned} \rho >2(b-a). \end{aligned}$$

(2.9)

Let us define the maps $\gamma ^\pm \in \mathrm{Lip}([a,b];\mathbb {R}^2)$ as follows: if $t \in [a,b]$,

$$\begin{aligned} \begin{aligned} \gamma ^-(t)&:=(1,0), \\ \gamma ^+(t)&:= \rho \left( \cos (\theta (t)), \sin (\theta (t))\right) + (1-\rho , 0), \end{aligned} \end{aligned}$$

(2.10)

where $\theta :[a,b]\rightarrow [0,2\pi ]$ is given by

$$\begin{aligned} \theta (t):=\frac{2\pi (t-a)}{b-a}, \qquad t\in [a,b], \end{aligned}$$

(2.11)

see the second picture of Fig. 2. Then $\varGamma :=\mathrm{graph}(\gamma ^-) \cup \mathrm{graph}(\gamma ^+)$ is a Lipschitz closed simple curve. Moreover, any disk-type surface spanning $\varGamma$ has area greater than or equal to the area $\pi \rho ^2$ of its orthogonal projection (a disk of radius $\rho$) on the coordinate plane $\mathbb {R}^2_{(\xi ,\eta )}$, hence

$$\begin{aligned} a(\varGamma ) \ge \pi \rho ^2. \end{aligned}$$

On the other hand, the image $\varPhi (\mathrm{R})$ of the semicartesian parametrization $(\mathrm{R},\varPhi )$ defined in (2.8) has area strictly less than $2 \pi \rho (b-a)$ (see the first picture in Fig. 2). From our choice (2.9), it then follows

$$\begin{aligned} m(\mathrm{R};\varGamma ^-, \varGamma ^+)< a(\varGamma ). \end{aligned}$$

Fig. 2

The curve $\varGamma$ defined in Example 2.13. The left picture is the image of the semicartesian parametrization $(\mathrm{R},\varPhi )$ spanning $\varGamma$ built as in Example 2.12; we notice that it lies on the lateral surface of the cylinder of base the disk of radius $\rho$ and height $b-a$. The right picture represents the image of an embedding of the disk mapping the boundary of the disk onto $\varGamma$: the area of such a surface is greater than or equal to the area of its orthogonal projection on a plane orthogonal to the t axis, that is a disk of radius $\rho$

We conclude this section proving that fixing $\mathrm{R}$ and D as in Definition 2.5 is not restrictive.

Lemma 2.14

(Choice of domain) Let $\varGamma = \varGamma ^- \cup \varGamma ^+$ be union of two Lipschitz graphs on [a, b], $\varGamma ^\pm = \mathrm{graph}(\gamma ^\pm )$. Let $O_1 = [[\sigma ^-_1, \sigma ^+_1]]$ be such that $\sigma ^-_1(a)<\sigma ^+_1(a)$ and $\sigma ^-_1(b)<\sigma ^+_1(b)$. If $(O_1, \varPsi )$ is a semicartesian parametrization spanning $\varGamma$ such that $\varPsi \in H^1(O_1; \mathbb {R}^3)$, then there exists a map $\varPhi \in \mathrm{semicart}(\mathrm{R};\varGamma ^-, \varGamma ^+)$ such that

$$\begin{aligned} \int _\mathrm{R}|\partial _t \varPhi \wedge \partial _s \varPhi |\, \text{ d }t \,\text{ d }s = \int _{O_1} |\partial _t \varPsi \wedge \partial _s \varPsi |\, dt\, ds. \end{aligned}$$

(2.12)

If moreover $\varGamma$ is closed, $O_2=[[\sigma ^+_2,\sigma ^+_2]]$ is such that $\sigma ^-_2(a)=\sigma ^+_2(a)$ and $\sigma ^-_2(b)=\sigma ^+_2(b)$, and $(O_2, \chi )$ is a semicartesian parametrization spanning $\varGamma$ such that $\chi \in H^1(O_2; \mathbb {R}^3)$, then there exists a map $\varPhi \in \mathrm{semicart}(D;\varGamma ^-, \varGamma ^+)$ such that

$$\begin{aligned} \int _D |\partial _t \varPhi \wedge \partial _s \varPhi |\,dt \,ds = \int _{O_2} |\partial _t \chi \wedge \partial _s \chi |\, dt\, ds. \end{aligned}$$

(2.13)

Proof

Let us define the map $T_1:\mathrm{R}\rightarrow O_1$ as

$$\begin{aligned} T_1(t,s):=\left( t, \frac{1-s}{2} \sigma ^-_1(t) + \frac{1+s}{2}\sigma ^+_1(t)\right) . \end{aligned}$$

Since $\sigma ^\pm _1 \in \mathrm{Lip}([a,b])$, we have that $T \in \mathrm{Lip}(\mathrm{R}; O_1)$ and thus the map $\varPhi :=\varPsi \circ T_1$ belongs to $\mathrm{semicart}(\mathrm{R};\varGamma ^-,\varGamma ^+)$; moreover $T_1$ is injective and thus (2.12) holds.

Let us suppose that $\varGamma$ is closed. Recall that $D=[[\sigma ^-, \sigma ^+]]$. We define the map $T_2:D\rightarrow O_2$ as

$$\begin{aligned} T_2(t,s):=\left( t, \frac{\sigma ^+(t)-s}{\sigma ^+(t)-\sigma ^-(t)} \sigma ^-_2(t) + \frac{s-\sigma ^-(t)}{\sigma ^+(t)-\sigma ^-(t)}\sigma ^+_2(t)\right) . \end{aligned}$$

One can show that $T_2 \in \mathrm{Lip}(D;O_2)$ with a computation similar to the one in Lemma 2.8, and thus the map $\varPhi :=\chi \circ T_2$ belongs to $\mathrm{semicart}(D;\varGamma ^-,\varGamma ^+)$. The injectivity of $T_2$ implies (2.13). $\square$

2.1 Maps from a planar domain to the plane and jumping on a curve

From now on we set

$$\begin{aligned} \mathrm{R}^-:= (a,b)\times (-1,0), \qquad \mathrm{R}^+:= (a,b)\times (0,1). \end{aligned}$$

Let $\Omega \subset \mathbb {R}^2_{(x,y)}$ be a bounded open connected set. Let ${\mathbf{u}}: \Omega \rightarrow \mathbb {R}^2_{(\xi ,\eta )}$ be a map belonging to $\mathrm{BV}(\Omega ; \mathbb {R}^2) \cap W^{1,\infty }(\Omega {\setminus } J_{\mathbf{u}};\mathbb {R}^2)$, where $J_{\mathbf{u}}\subset \Omega$ is a $\mathcal {C}^2$ simple curve parametrized by an arc-length parametrization $\alpha :(a,b)\subset \mathbb {R}_{t}\rightarrow \mathbb {R}^2_{(x,y)}$. Two cases are possible (remember our convention on the set $J_{\mathbf{u}}$ in the Introduction): either $J_{\mathbf{u}}\subset \subset \Omega$ or $J_{\mathbf{u}}\cap \partial \Omega \ne \emptyset$.

We denote by ${\mathbf{u}}^\pm$ the two Lipschitz traces on the two sides of the jump, and we define $\gamma ^\pm [{\mathbf{u}}] \in \mathrm{Lip}((a,b);\mathbb {R}^2)$ as

$$\begin{aligned} \gamma ^\pm [{\mathbf{u}}](t):={\mathbf{u}}^\pm (\alpha (t)). \end{aligned}$$

In accordance with our previous notation, we denote with $\varGamma ^\pm [{\mathbf{u}}] \subset \mathbb {R}^3=\mathbb {R}_t\times \mathbb {R}^2_{(\xi ,\eta )}$ the graph of $\gamma ^\pm [{\mathbf{u}}]$. When there is no ambiguity, we shall write $\gamma ^\pm$ and $\varGamma ^\pm$ in place of $\gamma ^\pm [{\mathbf{u}}]$ and $\varGamma ^\pm [{\mathbf{u}}]$, respectively.

In this paper, we will deal with pairs $(\Omega , {\mathbf{u}})$ satisfying one of the two conditions specified in Definitions 2.15 and 2.16. In both the two conditions, the jump $J_{\mathbf{u}}$ is a horizontal segment; this assumption allows to identify the plane $\mathbb {R}^2_{(x,y)}$ (containing the domain $\Omega$ of ${\mathbf{u}}$) with the space of the parameters $\mathbb {R}^2_{(t,s)}$, thus simplifying the presentation.⁶

Definition 2.15

(Condition I) We say that $\varOmega$ and ${\mathbf{u}}\in \mathrm{BV}(\varOmega ; \mathbb {R}^2)$ satisfy condition I if $\varOmega =\mathrm{R}$, $J_{\mathbf{u}}=(a,b)\times \{0\}$, and ${\mathbf{u}}\in \mathrm{Lip}(\mathrm{R}^-;\mathbb {R}^2)\cap \mathrm{Lip}(\mathrm{R}^+; \mathbb {R}^2)$.

Definition 2.16

(Condition II) We say that $\varOmega$ and ${\mathbf{u}}\in \mathrm{BV}(\varOmega ; \mathbb {R}^2)$ satisfy condition II if $J_{\mathbf{u}}:=[a,b]\times \{0\}\subset \subset \varOmega$, ${\mathbf{u}}\in W^{1,\infty }(\varOmega {\setminus } J_{\mathbf{u}};\mathbb {R}^2)$, and there exist the pointwise limits (still denoted by ${\mathbf{u}}^\pm$) of ${\mathbf{u}}$ at all points of $J_{\mathbf{u}}$.

3 Condition I: upper bound

The next proposition provides an upper bound for $\overline{{\mathcal {A}}}^\infty ({\mathbf{u}},\varOmega )$ (and hence for $\overline{{\mathcal {A}}}({\mathbf{u}},\varOmega )$), when $\varOmega$ and ${\mathbf{u}}$ satisfy condition I, proving one of the two inequalities (i.e., (3.2)) of Theorem 1.3. We shall suitably modify the construction made in [4] in a different context.

Proposition 3.1

(Upper bound, I) Let $\varOmega$ and ${\mathbf{u}}$ satisfy condition I. Then there exists a sequence $({\mathbf{u}}_h)\subset H^1 (\mathrm{R}; \mathbb {R}^2)$ converging to ${\mathbf{u}}$ in $L^1(\mathrm{R};\mathbb {R}^2)$ and uniformly out of $J_{\mathbf{u}}$ such that

$$\begin{aligned} \liminf _{h \rightarrow +\infty } {\mathcal {A}}({\mathbf{u}}_h, \mathrm{R}) = \int _\mathrm{R}|{\mathcal {M}}(\nabla {\mathbf{u}})|\,dt\,ds + m(\mathrm{R}; \varGamma ^-[{\mathbf{u}}],\varGamma ^+[{\mathbf{u}}]). \end{aligned}$$

(3.1)

Hence

$$\begin{aligned} \overline{{\mathcal {A}}}^\infty _s({\mathbf{u}}, \mathrm{R})\le m(\mathrm{R};\varGamma ^-[{\mathbf{u}}],\varGamma ^+[{\mathbf{u}}]). \end{aligned}$$

(3.2)

Proof

Let $(\varPhi _h)\subset \mathrm{semicart}(\mathrm{R};\varGamma ^-,\varGamma ^+)$ be a minimizing sequence for (1.3), that is

$$\begin{aligned} \int _\mathrm{R}|\partial _t\varPhi _h \wedge \partial _s \varPhi _h|\,dt\,ds\rightarrow m(\mathrm{R};\varGamma ^-,\varGamma ^+)\,\, \text{ as } h\rightarrow +\infty , \end{aligned}$$

(3.3)

and write $\varPhi _h(t,s)=(t, \phi _h(t,s))$ with $\phi _h \in H^1(\mathrm{R};\mathbb {R}^2)$. For any $\varepsilon \in (0,1)$ set $\mathrm{R}_\varepsilon := (a,b) \times (-\varepsilon ,\varepsilon )$, and define the map ${\mathbf{u}}_{h,\varepsilon }\in H^1(\mathrm{R}; \mathbb {R}^2)$ as

$$\begin{aligned} {\mathbf{u}}_{h,\varepsilon }(t,s):= {\left\{ \begin{array}{ll} {\mathbf{u}}(t,s) &{}\quad \mathrm{if}~ (t,s)\in \mathrm{R}{{\setminus }} \mathrm{R}_{2\varepsilon }, \\ {\mathbf{u}}(t,2(s-\varepsilon )) &{}\quad \mathrm{if}~ (t,s)\in (a,b)\times (\varepsilon ,2\varepsilon ), \\ {\mathbf{u}}(t,2(s+\varepsilon )) &{}\quad \mathrm{if}~ (t,s)\in (a,b)\times (-2\varepsilon ,-\varepsilon ), \\ \phi _h(t,s/\varepsilon ) &{}\quad \mathrm{if}~ (t,s)\in \overline{\mathrm{R}_{\varepsilon }}. \end{array}\right. } \end{aligned}$$

With a computation similar to the one in [4], we get

$$\begin{aligned} \liminf _{\varepsilon \rightarrow 0^+ } {\mathcal {A}}({\mathbf{u}}_{h,\varepsilon }, \mathrm{R})=\int _\mathrm{R}|{\mathcal {M}}(\nabla {\mathbf{u}})|\,\hbox {d}t\,\hbox {d}s + \int _{\mathrm{R}}\left| \partial _t \varPhi _h \wedge \partial _s \varPhi _h\right| \,\hbox {d}t\,\hbox {d}s. \end{aligned}$$

Indeed ${\mathcal {A}}({\mathbf{u}}_{h,\varepsilon }, (a,b)\times (\varepsilon ,2\varepsilon ))$ and ${\mathcal {A}}({\mathbf{u}}_{h,\varepsilon }, (a,b)\times (-2\varepsilon ,-\varepsilon ))$ are negligible as $\varepsilon \rightarrow 0^+$, as a consequence of the hypothesis ${\mathbf{u}}\in \mathrm{Lip}(\mathrm{R}^+; \mathbb {R}^2) \cap \mathrm{Lip}(\mathrm{R}^-;\mathbb {R}^2)$. Moreover, a direct computation gives:

$$\begin{aligned} \begin{aligned} {\mathcal {A}}({\mathbf{u}}_{h,\varepsilon },\mathrm{R}_\varepsilon )&=\int _a^b\int _{-\varepsilon }^\varepsilon \sqrt{1+ \left| \partial _t\phi _h\left( t, \frac{s}{\varepsilon }\right) \right| ^2 +\frac{1}{\varepsilon ^2}\left| \partial _s\phi _h\left( t, \frac{s}{\varepsilon }\right) \right| ^2 + \frac{1}{\varepsilon ^2}\left( \mathrm{\det } \nabla \phi _h \left( t,\frac{s}{\varepsilon }\right) \right) ^2} \hbox {d}s\,\hbox {d}t \\&=\int _a^b \int _{-1}^{1}\sqrt{\left| \partial _s\phi _h\left( t, s\right) \right| ^2 + \left( \mathrm{det} \nabla \phi _h \left( t,s\right) \right) ^2 + \mathcal {O}(\varepsilon ^2)}~\hbox {d}s\,\hbox {d}t \\&\,\,\,\,\overset{\varepsilon \rightarrow 0^+}{\longrightarrow } \int _{\mathrm{R}}\left| \partial _t \varPhi _h \wedge \partial _s \varPhi _h\right| ~\hbox {d}t\,\hbox {d}s. \end{aligned} \end{aligned}$$

By a diagonalization process, and using (3.3), we can choose a sequence $({\mathbf{u}}_{h}):=({\mathbf{u}}_{h,\varepsilon _h})$ such that

$$\begin{aligned} \lim _{h \rightarrow +\infty }{\mathcal {A}}({\mathbf{u}}_{h},\mathrm{R})=\int _\mathrm{R}|{\mathcal {M}}(\nabla {\mathbf{u}})|\,\hbox {d}t\,\hbox {d}s + m(\mathrm{R};\varGamma ^-,\varGamma ^+), \end{aligned}$$

which implies (3.1).⁷ $\square$

4 Condition I: lower bound

The main result of this section is the following inequality that, coupled with Proposition 3.1, concludes the proof of Theorem 1.3.

Theorem 4.1

(Lower bound, I) Let $\varOmega$ and ${\mathbf{u}}$ satisfy condition I. Let $({\mathbf{u}}_h) \subset \mathrm{Lip}(\mathrm{R}; \mathbb {R}^2)$ be a sequence converging to ${\mathbf{u}}$ in $L^1(\mathrm{R}; \mathbb {R}^2)$ and uniformly out of $J_{\mathbf{u}}$. Then

$$\begin{aligned} \liminf _{h\rightarrow +\infty } {\mathcal {A}}({\mathbf{u}}_h,\mathrm{R}) \ge \int _\mathrm{R}|{\mathcal {M}}(\nabla {\mathbf{u}})|\,\hbox {d}t\,\hbox {d}s+ m(\mathrm{R};\varGamma ^-[{\mathbf{u}}],\varGamma ^+[{\mathbf{u}}]). \end{aligned}$$

Hence

$$\begin{aligned} \overline{{\mathcal {A}}}^\infty _s({\mathbf{u}},\mathrm{R})\ge m(\mathrm{R};\varGamma ^-[{\mathbf{u}}],\varGamma ^+[{\mathbf{u}}]). \end{aligned}$$

(4.1)

The proof of Theorem 4.1 will be achieved in two steps: the first step gives the result if the sequence ${\mathbf{u}}_h$ coincides with ${\mathbf{u}}$ far enough from $J_{\mathbf{u}}$.

Theorem 4.2

Let $\varOmega$ and ${\mathbf{u}}$ satisfy condition I. If $({\mathbf{u}}_h)\subset \mathrm{Lip}(\mathrm{R};\mathbb {R}^2)$ converges to ${\mathbf{u}}$ in $L^1(\mathrm{R};\mathbb {R}^2)$ and

$$\begin{aligned} {\mathbf{u}}_h={\mathbf{u}}\qquad \text{ in } \mathrm{R}{\setminus } N_{h}, \end{aligned}$$

for some decreasing sequence $(N_h)$ of neighborhoods of $J_{\mathbf{u}}$ such that $\displaystyle \bigcap _{h \in \mathbb N} N_h = J_{\mathbf{u}}$, then

$$\begin{aligned} \liminf _{h\rightarrow +\infty }{\mathcal {A}}({\mathbf{u}}_h,\mathrm{R})\ge \int _\mathrm{R}|{\mathcal {M}}(\nabla {\mathbf{u}})|\,dt\,ds+ m(\mathrm{R};\varGamma ^-[{\mathbf{u}}],\varGamma ^+[{\mathbf{u}}]). \end{aligned}$$

The second step shows that, given any sequence $({\mathbf{u}}_h)$ satisfying the hypotheses of Theorem 4.1, we can build a sequence $({\mathbf{v}}_h)$ satisfying the hypotheses of Theorem 4.2 and whose area is, in the limit, not larger than the area of $({\mathbf{u}}_h)$.

Theorem 4.3

Let $\varOmega$ and ${\mathbf{u}}$ satisfy condition I. Let $({\mathbf{u}}_h) \subset \mathrm{Lip}(\mathrm{R}; \mathbb {R}^2)$ be a sequence converging to ${\mathbf{u}}$ in $L^1(\mathrm{R}; \mathbb {R}^2)$ and uniformly out of $J_{\mathbf{u}}$. Then there exists a sequence $({\mathbf{v}}_{h})\subset \mathrm{Lip}(\mathrm{R};\mathbb {R}^2)$ satisfying the hypotheses of Theorem 4.2 and such that

$$\begin{aligned} \liminf _{h\rightarrow +\infty } {\mathcal {A}}({\mathbf{u}}_h,\mathrm{R})\ge \liminf _{h\rightarrow +\infty } {\mathcal {A}}({\mathbf{v}}_h,\mathrm{R}). \end{aligned}$$

(4.2)

4.1 Proof of Theorem 4.2

We need some preliminary lemmas.

Lemma 4.4

(Lower bound of m via interpolation) Let $\alpha , \beta \in \mathrm{Lip}([a,b]; \mathbb {R}^2)$, and set $\varGamma _\alpha :=\mathrm{graph}(\alpha )$ and $\varGamma _\beta :=\mathrm{graph}(\beta )$. Then there exists a constant $C>0$ independent of $\alpha$ and $\beta$, such that

$$\begin{aligned} m(\mathrm{R}; \varGamma _\alpha ,\varGamma _\beta )\le C||\alpha -\beta ||_{L^1((a,b); \mathbb {R}^2)} \left( 1+\max \left[ ||\dot{\alpha }||_{L^\infty ((a,b);\mathbb {R}^2)},||\dot{\beta }||_{L^\infty ((a,b);\mathbb {R}^2)}\right] \right) . \end{aligned}$$

(4.3)

Proof

Let us define the map $\ell \in W^{1,\infty }(\mathrm{R}; \mathbb {R}^2)$ interpolating $\alpha$ and $\beta$, as in Lemma 2.8, that is

$$\begin{aligned} \ell (t,s):= \frac{1-s}{2}\alpha (t)+ \frac{1+s}{2}\beta (t), \qquad (t,s) \in \mathrm{R}. \end{aligned}$$

Setting $\varPhi _\ell (t,s):=(t,\ell (t,s))$, we get $\partial _t \varPhi _\ell (t,s)=\left( 1,\frac{1-s}{2}\dot{\alpha }(t) +\frac{1+s}{2}\dot{\beta }(t) \right)$ and $\partial _s \varPhi _\ell (t,s)=\left( 0,\frac{\beta (t)-\alpha (t)}{2}\right)$. Thus

$$\begin{aligned} \left| \partial _t \varPhi _\ell \wedge \partial _s\varPhi _\ell \right| =\frac{1}{2}\sqrt{|\alpha -\beta |^2+\left[ \left( \frac{1-s}{2}\dot{\alpha }+\frac{1+s}{2}\dot{\beta }\right) \cdot (\alpha -\beta )^\perp \right] ^2}, \end{aligned}$$

(4.4)

where, for $z=(z_1,z_2) \in \mathbb {R}^2$, we set $z^\perp :=(-z_2,z_1)$. Hence

$$\begin{aligned}&\int _\mathrm{R}\left| \partial _t \varPhi _\ell \wedge \partial _s\varPhi _\ell \right| \,dt\,ds\nonumber \\&\quad \le C||\alpha -\beta ||_{L^1((a,b); \mathbb {R}^2)}\left( 1+ \max \left[ ||\dot{\alpha }||_{L^\infty ((a,b);\mathbb {R}^2)},||\dot{\beta }||_{L^\infty ((a,b);\mathbb {R}^2)}\right] \right) , \end{aligned}$$

(4.5)

and, since $\varPhi _\ell \in \mathrm{semicart}(\mathrm{R};\varGamma _\alpha ,\varGamma _\beta )$, also (4.3) follows. $\square$

The computations in Lemma 4.4 allow to prove a semicontinuity result for $m(\mathrm{R};\varGamma ^-_h, \varGamma ^+_h)$.

Lemma 4.5

(Lower semicontinuity of $m(\mathrm{R}; \cdot , \cdot )$) Let $(\gamma ^\pm _h)\subset \mathrm{Lip}([a,b];\mathbb {R}^2)$ and $\gamma ^\pm \in \mathrm{Lip}([a,b]; \mathbb {R}^2)$ be such that:

• there exists $C_1>0$ such that $||\dot{\gamma }^\pm _h||_{L^\infty ((a,b);\mathbb {R}^2)} \le C_1$ for any $h \in \mathbb N$,

• $\gamma ^\pm _h\rightarrow \gamma ^\pm$ in $L^1((a,b);\mathbb {R}^2)$ as $h \rightarrow +\infty$.

Then, setting $\varGamma ^\pm _h := \mathrm{graph}(\gamma _h^\pm )$ and $\varGamma ^\pm := \mathrm{graph}(\gamma ^\pm )$, we have

$$\begin{aligned} m(\mathrm{R}; \varGamma ^-, \varGamma ^+)\le \liminf _{h\rightarrow +\infty } m(\mathrm{R}; \varGamma ^-_h, \varGamma ^+_h). \end{aligned}$$

Proof

For any $h \in \mathbb N$, let $\left( \varPhi ^h_k\right) \subset \mathrm{semicart}(\mathrm{R};\varGamma ^-_h,\varGamma ^+_h)$ be such that

$$\begin{aligned} \lim _{k\rightarrow +\infty }\int _{\mathrm{R}}|\partial _t \varPhi ^h_k\wedge \partial _s\varPhi ^h_k|\, dt\, ds = m(\mathrm{R}; \varGamma ^-_h, \varGamma ^+_h). \end{aligned}$$

(4.6)

Let us denote by $\ell ^+_h, \ell ^-_h: \mathrm{R}\rightarrow \mathbb {R}^2$ the linear interpolating maps, such that, for any $t \in [a,b]$,

$$\begin{aligned} \begin{aligned} \ell ^+_h(t,-1)= \gamma ^+_h(t),&\qquad \ell ^+_h(t,1)= \gamma ^+(t), \\ \ell ^-_h(t,-1)= \gamma ^-_h(t),&\qquad \ell ^-_h(t,1)= \gamma ^-(t). \end{aligned} \end{aligned}$$

Following the notation of Lemma 2.8 we also write $\varPhi _{\ell ^\pm _h}(t,s):=(t, \ell ^\pm _h(t,s))$. We define the maps $(\varPsi ^h_k)\subset \mathrm{semicart}(\mathrm{R}; \varGamma ^-,\varGamma ^+)$ as

$$\begin{aligned} \varPsi ^h_k(t,s):= {\left\{ \begin{array}{ll} \varPhi _{\ell ^+_h}(t, 4s-3) \quad &{} \quad \text{ if } t \in (a,b), s\in [1/2, 1), \\ \varPhi ^h_k(t, 2s) &{}\quad \text{ if } t \in (a,b), s\in (-1/2, 1/2),\\ \varPhi _{\ell ^-_h}(t, -4s-3) &{}\quad \text{ if } t \in (a,b), s\in (-1, -1/2].\\ \end{array}\right. } \end{aligned}$$

We have, using also (4.6),

$$\begin{aligned} \begin{aligned}&\int _\mathrm{R}|\partial _t \varPsi ^h_k \wedge \partial _s \varPsi ^h_k|\, \hbox {d}t \, \hbox {d}s \\&=\int _\mathrm{R}|\partial _t \varPhi ^h_k \wedge \partial _s \varPhi ^h_k| \, \hbox {d}t \, \hbox {d}s + \int _\mathrm{R}|\partial _t \varPhi _{\ell ^+_h} \wedge \partial _s \varPhi _{\ell ^+_h}|\, \hbox {d}t \, \hbox {d}s + \int _\mathrm{R}|\partial _t \varPhi _{\ell ^-_h} \wedge \partial _s \varPhi _{\ell ^-_h}|\, \hbox {d}t \, \hbox {d}s \\&\quad \overset{\tiny {k\rightarrow +\infty }}{\longrightarrow } m(\mathrm{R}; \varGamma ^-_h,\varGamma ^+_h) + \int _\mathrm{R}|\partial _t \varPhi _{\ell ^+_h} \wedge \partial _s \varPhi _{\ell ^+_h}|\, \hbox {d}t \, \hbox {d}s + \int _\mathrm{R}|\partial _t \varPhi _{\ell ^-_h} \wedge \partial _s \varPhi _{\ell ^-_h}|\, \hbox {d}t \, \hbox {d}s. \end{aligned} \end{aligned}$$

Now, recalling inequality (4.5) and our first assumption, we have

$$\begin{aligned} \begin{aligned}&\int _\mathrm{R}|\partial _t \varPhi _{\ell ^+_h} \wedge \partial _s \varPhi _{\ell ^+_h}|\, \text{ d }t \, \text{ d }s + \int _\mathrm{R}|\partial _t \varPhi _{\ell ^-_h} \wedge \partial _s \varPhi _{\ell ^-_h}|\, \text{ d }t \, \text{ d }s \\&\quad \le C \left( \Vert \gamma _h^+ - \gamma ^+\Vert _{L^1((a,b); \mathbb {R}^2)} +\Vert \gamma _h^- - \gamma ^-\Vert _{L^1((a,b); \mathbb {R}^2)} \right) \left( 1 + C_1 \right) , \end{aligned} \end{aligned}$$

and the right hand side is infinitesimal as $h \rightarrow +\infty$ by our second assumption. Hence, we can select a subsequence $(k_h)$ and obtain a sequence $\left( \varPsi ^h_{k_h}\right) \subset \mathrm{semicart}(\mathrm{R}; \varGamma ^-, \varGamma ^+)$ so that

$$\begin{aligned} \liminf _{h \rightarrow +\infty } \int _{\mathrm{R}} |\partial _t \varPsi ^h_{k_h} \wedge \partial _s \varPsi ^h_{k_h} |\,\hbox {d}t\,\hbox {d}s = \liminf _{h \rightarrow +\infty } m(\mathrm{R}; \varGamma ^-_h,\varGamma ^+_h). \end{aligned}$$

The inclusion $\varPsi ^h_{k_h} \in \mathrm{semicart}(\mathrm{R}; \varGamma ^-,\varGamma ^+)$ implies that

$$\begin{aligned} \int _{\mathrm{R}} |\partial _t \varPsi ^h_{k_h} \wedge \partial _s \varPsi ^h_{k_h} |\,\hbox {d}t\,\hbox {d}s \ge m(R; \varGamma ^-, \varGamma ^+), \end{aligned}$$

and the assertion of the lemma follows. $\square$

The last result that we need before proving Theorem 4.2 provides an estimate from below of the area of the graph of a sufficiently smooth map on a strip.

Lemma 4.6

(Lower bound of area on a strip) Let $\varepsilon \in (0,1)$ and $\mathrm{R}_\varepsilon :=(a,b)\times (-\varepsilon , \varepsilon )$. Given a map ${\mathbf{v}}\in \mathrm{Lip}(\mathrm{R}_\varepsilon ;\mathbb {R}^2)$, let $\varGamma ^\pm _\varepsilon$ denote the graphs on [a, b] of the sections ${\mathbf{v}}(\cdot , \pm \varepsilon )\in \mathrm{Lip}([a,b];\mathbb {R}^2)$. Then

$$\begin{aligned} {\mathcal {A}}({\mathbf{v}}, \mathrm{R}_\varepsilon )\ge m(\mathrm{R}; \varGamma ^-_\varepsilon , \varGamma ^+_\varepsilon ). \end{aligned}$$

Proof

Set ${\mathbf{v}}= (v_1,v_2)$. Neglecting the constant 1 and the term $|\partial _t {\mathbf{v}}|^2$ in the expression of $|{\mathcal {M}}(\nabla {\mathbf{v}})|$, we deduce

$$\begin{aligned} {\mathcal {A}}({\mathbf{v}}, \mathrm{R}_\varepsilon )\ge \int _{\mathrm{R}_\varepsilon }\sqrt{|\partial _s {\mathbf{v}}|^2 + (\partial _t v_{1}\partial _s v_{2}-\partial _s v_{1} \partial _t v_{2})^2}~\hbox {d}t\,\hbox {d}s. \end{aligned}$$

(4.7)

On the other hand we can define the map $\varPhi \in \mathrm{semicart}(\mathrm{R}; \varGamma ^-_\varepsilon , \varGamma ^+_\varepsilon )$ as

$$\begin{aligned} \varPhi (t,s):=(t, \mathbf{v}(t,\varepsilon s)), \qquad (t,s) \in \mathrm{R}, \end{aligned}$$

and (2.2) shows that $\int _\mathrm{R}|\partial _t \varPhi \wedge \partial _s \varPhi |\,\hbox {d}t\,\hbox {d}s$ equals the right hand side of (4.7). Hence

$$\begin{aligned} {\mathcal {A}}({\mathbf{v}}, \mathrm{R}_\varepsilon )\ge \int _\mathrm{R}|\partial _t \varPhi \wedge \partial _s \varPhi |\,\hbox {d}t\,\hbox {d}s \ge m(\mathrm{R}; \varGamma ^-_\varepsilon , \varGamma ^+_\varepsilon ). \end{aligned}$$

$\square$

Now, we can prove Theorem 4.2.

Proof

Recalling the properties of the sequence $({\mathbf{u}}_{h})$, we can choose an infinitesimal sequence $(\varepsilon _h)$ of positive numbers such that $\mathrm{R}_{\varepsilon _h}:=(a,b)\times (-\varepsilon _h, \varepsilon _h)\supseteq N_h$. We have

$$\begin{aligned} {\mathcal {A}}({\mathbf{u}}_h, \mathrm{R})={\mathcal {A}}({\mathbf{u}}, \mathrm{R}{\setminus } \mathrm{R}_{\varepsilon _h}) + {\mathcal {A}}({\mathbf{u}}_{h}, \mathrm{R}_{\varepsilon _h}). \end{aligned}$$

Set $\gamma _{h}^\pm (\cdot ):={\mathbf{u}}_h(\cdot ,\pm \varepsilon _h)$ and $\gamma ^\pm :=\gamma ^\pm [{\mathbf{u}}]$. We observe that, by assumption, $\gamma ^\pm _h={\mathbf{u}}(\cdot ,\pm \varepsilon _h)$ and thus $\gamma ^\pm _h$ and $\gamma$ satisfy the hypotheses of Lemma 4.5. Hence, applying also Lemma 4.6, we get

$$\begin{aligned} \liminf _{h\rightarrow +\infty } {\mathcal {A}}({\mathbf{u}}_h, \mathrm{R})&\ge \liminf _{h\rightarrow +\infty }\left[ {\mathcal {A}}({\mathbf{u}}, \mathrm{R}{\setminus } \mathrm{R}_{\varepsilon _h}) + m(\mathrm{R}; \varGamma ^-_h, \varGamma ^+_h)\right] \\&\ge \int _\mathrm{R}|{\mathcal {M}}(\nabla {\mathbf{u}})|\,\hbox {d}t\,\hbox {d}s + m(\mathrm{R}; \varGamma ^-,\varGamma ^+), \end{aligned}$$

that is the thesis. $\square$

Remark 4.7

The strategy of the proof of Theorem 4.2 would prove the lower bound (4.1) for any sequence $({\mathbf{u}}_h)\subset {\mathcal {C}}^1(\mathrm{R}; \mathbb {R}^2)$ converging to ${\mathbf{u}}$ in $L^1(\mathrm{R}; \mathbb {R}^2)$, if we would be able to remove the bound on the $L^\infty$-norm of $\dot{\gamma }^\pm _h$ in the hypotheses of Lemma 4.5. Indeed, as a consequence of Fubini’s theorem, the convergence of $({\mathbf{u}}_h)$ to ${\mathbf{u}}$ in $L^1(\mathrm{R}; \mathbb {R}^2)$ implies that ${\mathbf{u}}_h(\cdot ,\varepsilon )\rightarrow {\mathbf{u}}(\cdot , \varepsilon )$ in $L^1((a,b);\mathbb {R}^2)$ for almost every level $\varepsilon \in (0,1)$.

Lemma 4.5 is in some sense coherent with the lower semicontinuity of the area of solutions of Plateau’s problem (when $\varGamma _h$ and $\varGamma$ are Jordan curves); indeed lower semicontinuity is usually guaranteed when $\varGamma _h \rightarrow \varGamma$ in the sense of Fréchet, [8, 13], that would be implied by our hypotheses. On the other hand, in Example 4.8 we exhibit a sequence $(\varGamma _h)$ of curves, union of two Lipschitz graphs, converging in $L^1$ to a union $\varGamma$ of two Lipschitz graphs, for which the lower semicontinuity fails. In this context $\varGamma _h \rightarrow \varGamma$ in $L^1$ means that $\varGamma _h:=\mathrm{graph}(\gamma ^-_h)\cup \mathrm{graph}(\gamma ^+_h)$, $\varGamma :=\mathrm{graph}(\gamma ^-)\cup \mathrm{graph}(\gamma ^+)$, $\gamma ^\pm _h,\,\gamma ^\pm \in \mathrm{Lip}([a,b]; \mathbb {R}^2)$, and $\gamma ^\pm _h \rightarrow \gamma ^\pm$ in $L^1((a,b);\mathbb {R}^2)$ as $h \rightarrow +\infty$.

Example 4.8

(Lack of $L^1$-lower semicontinuity for the Plateau’s problem) Let $\varGamma \subset \mathbb {R}^3$ be a closed simple rectifiable curve. As already recalled in Remark 4.7, it is known that $a(\cdot )$ is Fréchet lower semicontinuous. We show here that if $\varGamma$ is union of the graphs of $\gamma ^\pm \in \mathrm{Lip}([a,b];\mathbb {R}^2)$, it may happen that

$$\begin{aligned} a(\varGamma ) > \liminf _{h\rightarrow +\infty } a(\varGamma _h), \end{aligned}$$

(4.8)

where $(\varGamma _h)$ is a sequence of closed simple space curves, $\varGamma _h = \mathrm{graph}(\gamma _h^-) \cup \mathrm{graph}(\gamma ^+_h)$, and $\gamma _h^\pm \rightarrow \gamma ^\pm$ in $L^1((a,b); \mathbb {R}^2)$ as $h \rightarrow +\infty$.

Indeed, choose the maps $\gamma ^\pm$ as in (2.10). Let us define the maps $\gamma ^\pm _h\in \mathrm{Lip}([a,b];\mathbb {R}^2)$ converging to $\gamma ^\pm$ in $L^1((a,b);\mathbb {R}^2)$ as

$$\begin{aligned} \gamma ^-_h(t)=(1,0)=\gamma ^-(t), \qquad \qquad \gamma ^+_h(t)= \rho \big (\cos (\theta _h(t)), \sin (\theta _h(t))\big ) + (1-\rho , 0), \end{aligned}$$

where

$$\begin{aligned} \theta _h(t):= {\left\{ \begin{array}{ll} \theta (t) \, &{} \mathrm{if}~t\in [a,b-h^{-1}], \\ -2\pi \frac{b-a - h^{-1}}{b-a} h (t-b) \, &{} \mathrm{if}~t \in (b-h^{-1},b], \end{array}\right. } \end{aligned}$$

with $\theta$ defined in (2.11). Hence, in the short interval $(b-h^{-1}, b)$, the path made by $\gamma ^+_h$ is the same as the path it makes in $(a, b-h^{-1})$, with reversed orientation. The curve $\varGamma _h$ is represented in Fig. 3. For any $h \in \mathbb N$ there exists an immersion of the disk, mapping the boundary of the disk onto $\varGamma _h$ whose image lies on the lateral boundary of the cylinder $[a,b]\times B_\rho ((1-\rho ,0))$; hence for any $h\in \mathbb N$ we have $a(\varGamma _h)\le 2\pi \rho (b-a)$ that, for $\rho$ large enough, gives (4.8), see Example 2.13.

This example does not exclude the lower semicontinuity of $m(\mathrm{R};\varGamma ^-,\varGamma ^+)$ with respect to the $L^1$-convergence. Indeed the limit of the areas of the surfaces represented in Fig. 3 is the area of the surface represented in the first picture of Fig. 2.

Fig. 3

One element of the sequence $(\varGamma _h)$ defined in Example 4.8 that approximates in $L^1$ sense the curve $\varGamma$ defined in Example 2.13, and a disk-type surface with boundary $\varGamma _h$ that lies on the cylinder

4.2 Proof of Theorem 4.3

In order to prove Theorem 4.3 we need the following technical result, inspired by [1, Proposition 7.3], that provides a way to interpolate two maps on a strip, by controlling the amount of area of the interpolating map with the thickness of the strip.

Proposition 4.9

(Interpolation, I) Let $({\mathbf{u}}_h)\subset \mathrm{Lip}(\mathrm{R}^+;\mathbb {R}^2)$ be a sequence converging to ${\mathbf{u}}\in \mathrm{Lip}(\mathrm{R}^+;\mathbb {R}^2)$ in $L^1(\mathrm{R}^+;\mathbb {R}^2)$. Let $\varepsilon _o \in (0,1)$ be fixed, such that $\partial _t {\mathbf{u}}(\cdot , s)_{|s=\varepsilon _o}$ exists almost everywhere in (a, b). Let $\varepsilon _i \in (0,\varepsilon _o)$ be such that:

• (i) $||{\mathbf{u}}_h(\cdot , \varepsilon _i) - {\mathbf{u}}(\cdot , \varepsilon _i)||_{L^\infty ((a,b);\mathbb {R}^2)}\rightarrow 0$ as $h \rightarrow +\infty$;

• (ii) $\partial _t {\mathbf{u}}_h(\cdot , s)_{|s=\varepsilon _i}$ exists almost everywhere in (a, b) for any $h \in \mathbb {N}$;

• (iii) $\displaystyle \liminf _{h\rightarrow +\infty }||\partial _t {\mathbf{u}}_h(\cdot , \varepsilon _i)||_{L^1((a,b);\mathbb {R}^2)}\le M$, where the constant M may depend on $\varepsilon _i$.

Then the sequence $({\mathbf{v}}_h)\subset \mathrm{Lip}(\mathrm{R}^+;\mathbb {R}^2)$ defined as

$$\begin{aligned} {\mathbf{v}}_h(t,s):= {\left\{ \begin{array}{ll} {\mathbf{u}}(t,s) &{}\quad \mathrm{if}~ t\in (a,b)\,, s>\varepsilon _o , \\ \frac{\varepsilon _o - s}{\varepsilon _o - \varepsilon _i}{\mathbf{u}}_h (t,\varepsilon _i)+\frac{s- \varepsilon _i}{\varepsilon _o - \varepsilon _i}{\mathbf{u}}(t,\varepsilon _o) \, &{}\quad \mathrm{if}~ t\in (a,b),\,s \in [\varepsilon _i,\varepsilon _o], \\ {\mathbf{u}}_h(t,s) &{}\quad \mathrm{if}~ t\in (a,b)\, , s<\varepsilon _i \end{array}\right. } \end{aligned}$$

(4.9)

satisfies

$$\begin{aligned} \liminf _{h\rightarrow +\infty } {\mathcal {A}}\big ({\mathbf{v}}_{h}, (a,b)\times (\varepsilon _i, \varepsilon _o)\big )\le C\big [1 + M\big ]|\varepsilon _o-\varepsilon _i|, \end{aligned}$$

(4.10)

where $C>0$ depends on $\mathrm{lip}({\mathbf{u}})$ and $b-a$, and is independent of $\varepsilon _o$ and $\varepsilon _i$.

Proof

Let $S_{\varepsilon _i}^{\varepsilon _o} := (a,b)\times (\varepsilon _i, \varepsilon _o)$. The Jacobian matrix of ${\mathbf{v}}_h$ at almost every $(t,s)\in S_{\varepsilon _i}^{\varepsilon _o}$ is:

$$\begin{aligned} \frac{1}{\varepsilon _o-\varepsilon _i} \left( (\varepsilon _o-s)\partial _t{\mathbf{u}}_{h}(t,\varepsilon _i) +(s-\varepsilon _i)\partial _t{\mathbf{u}}(t,\varepsilon _o)\,\, \Bigg \vert \,\, {\mathbf{u}}(t,\varepsilon _o)-{\mathbf{u}}_{h} (t,\varepsilon _i) \right) . \end{aligned}$$

We control the area of the graph of ${\mathbf{v}}_{h}$ in $S_{\varepsilon _i}^{\varepsilon _o}$ as

$$\begin{aligned} {\mathcal {A}}({\mathbf{v}}_{h},S_{\varepsilon _i}^{\varepsilon _o}) \le C\int _{S_{\varepsilon _i}^{\varepsilon _o}} \Big [[1 + |\partial _t {\mathbf{v}}_{h}|+|\partial _s {\mathbf{v}}_{h}|+|\mathrm{det}\nabla {\mathbf{v}}_{h}|\Big ]\, \hbox {d}t \, \hbox {d}s, \end{aligned}$$

(4.11)

where $C>0$ is an absolute constant. We estimate each of the four integrals on the right hand side of (4.11) as follows.

• The first term is obviously bounded by $(\varepsilon _o-\varepsilon _i)(b-a)$.

• Concerning the second term, we have

  $$\begin{aligned} \int _{\varepsilon _i}^{\varepsilon _o}\int _a^b |\partial _t{\mathbf{v}}_{h}|\,\hbox {d}t\,\hbox {d}s\le & {} (\varepsilon _o-\varepsilon _i) \int _a^b \left( |\partial _t{\mathbf{u}}_{h}(t,\varepsilon _i)|+|\partial _t{\mathbf{u}}(t,\varepsilon _o)|\right) \,\hbox {d}t\nonumber \\\le & {} (\varepsilon _o - \varepsilon _i) \left[ \int _a^b |\partial _t{\mathbf{u}}_{h}(t,\varepsilon _i)|\,\hbox {d}t + \mathrm{lip}({\mathbf{u}})(b-a) \right] . \end{aligned}$$

  (4.12)

• Similarly, for the third term we have

  $$\begin{aligned} \begin{aligned} \int _{\varepsilon _i}^{\varepsilon _o}\int _a^b |\partial _s{\mathbf{v}}_{h}(t,s)|\,\hbox {d}t\,\hbox {d}s\le&\int _a^b\Big [ |{\mathbf{u}}(t,\varepsilon _o)- {\mathbf{u}}(t,\varepsilon _i)| + |{\mathbf{u}}(t,\varepsilon _i)-{\mathbf{u}}_{h}(t,\varepsilon _i)|\Big ]\,\hbox {d}t\\ \le&\, \mathrm{lip}({\mathbf{u}})(b-a)(\varepsilon _o-\varepsilon _i) + ||{\mathbf{u}}(\cdot ,\varepsilon _i)- {\mathbf{u}}_{h}(\cdot ,\varepsilon _i)||_{L^1((a,b);\mathbb {R}^2)}. \end{aligned} \end{aligned}$$

  (4.13)

• Concerning the term with the determinant:

  $$\begin{aligned} \begin{aligned}&\int _{\varepsilon _i}^{\varepsilon _o}\int _a^b |\det \nabla {\mathbf{v}}_{h}(t,s)|~\hbox {d}t\hbox {d}s\\&\quad \le 2\int _a^b |{\mathbf{u}}(t,\varepsilon _o)- {\mathbf{u}}_{h}(t,\varepsilon _i)|\big (|\partial _t{\mathbf{u}}_{h}(t,\varepsilon _i)|+|\partial _t{\mathbf{u}}(t,\varepsilon _o)|\big )\,\hbox {d}t\\&\quad = 2 \int _a^b |{\mathbf{u}}(t,\varepsilon _o)- {\mathbf{u}}_{h}(t,\varepsilon _i)||\partial _t{\mathbf{u}}(t,\varepsilon _o)|\,\hbox {d}t + 2 \int _a^b |{\mathbf{u}}(t,\varepsilon _o)- {\mathbf{u}}_{h}(t,\varepsilon _i)||\partial _t{\mathbf{u}}_{h}(t,\varepsilon _i)|\,\hbox {d}t \\&\quad =: \mathrm{I}_{h} + \mathrm{II}_{h}. \end{aligned} \end{aligned}$$

  We have

  $$\begin{aligned} \begin{aligned} \mathrm{I}_{h} \le&2{\mathrm{lip}}({\mathbf{u}}) \int _a^b \left( |{\mathbf{u}}(t,\varepsilon _o)-{\mathbf{u}}(t,\varepsilon _i) \vert + \vert {\mathbf{u}}(t,\varepsilon _i) - {\mathbf{u}}_{h}(t,\varepsilon _i)|\right) \,\hbox {d}t\\ \le&2 ({\mathrm{lip}}({\mathbf{u}}))^2(b-a) (\varepsilon _o-\varepsilon _i) + 2 {\mathrm{lip}}({\mathbf{u}}) ||{\mathbf{u}}(\cdot ,\varepsilon _i)-{\mathbf{u}}_{h}(\cdot ,\varepsilon _i)||_{L^1((a,b);\mathbb {R}^2)}. \end{aligned} \end{aligned}$$

  (4.14)

  Next

  $$\begin{aligned} \begin{aligned} \mathrm{II}_{h}&\le 2 \int _a^b |{\mathbf{u}}(t,\varepsilon _o) - {\mathbf{u}}(t,\varepsilon _i)||\partial _t {\mathbf{u}}_{h}(t,\varepsilon _i)|\,\hbox {d}t + 2 \int _a^b |{\mathbf{u}}(t,\varepsilon _i) - {\mathbf{u}}_{h}(t,\varepsilon _i)||\partial _t{\mathbf{u}}_{h}(t,\varepsilon _i)|\,\hbox {d}t\\&\le 2 \big (\mathrm{lip}({\mathbf{u}})(\varepsilon _o - \varepsilon _i)+ ||{\mathbf{u}}(\cdot ,\varepsilon _i)-{\mathbf{u}}_{h}(\cdot ,\varepsilon _i)||_{L^\infty ((a,b);\mathbb {R}^2)}\big )\int _a^b|\partial _t {\mathbf{u}}_h (t,\varepsilon _i)|\, \hbox {d}t. \end{aligned} \end{aligned}$$

  (4.15)

Finally, using (4.12), (4.13), (4.14) and (4.15) we get:

$$\begin{aligned} \begin{aligned} {\mathcal {A}}({\mathbf{v}}_h, S_{\varepsilon _i}^{\varepsilon _o})&\le C(\varepsilon _o - \varepsilon _i) \Big [ 1 + ||{\mathbf{u}}(\cdot , \varepsilon _i)-{\mathbf{u}}_h(\cdot , \varepsilon _i)||_{L^1((a,b);\mathbb {R}^2)} \\&\quad +(1 + ||{\mathbf{u}}(\cdot , \varepsilon _i)-{\mathbf{u}}_h(\cdot , \varepsilon _i)||_{L^\infty ((a,b);\mathbb {R}^2)})\int _a^b |\partial _t {\mathbf{u}}_h(t,\varepsilon _i)|\,\hbox {d}t \Big ]. \end{aligned} \end{aligned}$$

Using hypotheses (i)–(iii), and passing to the limit, we get

$$\begin{aligned} \liminf _{h\rightarrow +\infty } {\mathcal {A}}\left( {\mathbf{v}}_{h}, S_{\varepsilon _i}^{\varepsilon _o}\right) \le C [1+ M](\varepsilon _o-\varepsilon _i), \end{aligned}$$

(4.16)

where C depends just on $b-a$ and $\mathrm{lip}({\mathbf{u}})$. $\square$

We are now in the position to prove Theorem 4.3.

Proof

We can suppose that ${\mathcal {A}}({\mathbf{u}}_h,\mathrm{R})$ is uniformly bounded with respect to $h \in \mathbb N$, otherwise the result is trivial. Moreover, passing to a not relabeled subsequence, we can suppose also that there exist

$$\begin{aligned} \lim _{h\rightarrow +\infty } {\mathcal {A}}({\mathbf{u}}_h,\mathrm{R}) < + \infty , \qquad \lim _{h\rightarrow +\infty } {\mathcal {A}}({\mathbf{u}}_h,\mathrm{R}^+)< + \infty , \qquad \lim _{h\rightarrow +\infty } {\mathcal {A}}({\mathbf{u}}_h,\mathrm{R}^-)< + \infty . \end{aligned}$$

Since ${\mathbf{u}}_h\rightarrow {\mathbf{u}}$ uniformly on every compact set of $\mathrm{R}^+$ as $h \rightarrow +\infty$, hypothesis (i) of Proposition 4.9 is verified for any choice of the level $\varepsilon _1 \in (0,1)$. Using Fatou’s lemma we get

$$\begin{aligned} \int _0^1 \liminf _{h \rightarrow +\infty } \left( \int _a^b |\partial _t{\mathbf{u}}_{h}(t, s)|\,\text{ d }t\right) \, \text{ d }s\le \liminf _{h\rightarrow +\infty } {\mathcal {A}}({\mathbf{u}}_h, \mathrm{R}^+)< +\infty . \end{aligned}$$

Thus we can select a level $\varepsilon _1 \in (0,1)$, a subsequence $({\mathbf{u}}_{h_j})$ and a constant $M(\varepsilon _1)$ both depending on $\varepsilon _1$, such that

$$\begin{aligned} \lim _{j \rightarrow +\infty } \int _a^b |\partial _t{\mathbf{u}}_{h_j}(t, \varepsilon _1)|\,\text{ d }t\le M(\varepsilon _1). \end{aligned}$$

Repeating the argument a countably number of times and using the same procedure on $\mathrm{R}^-$, we can select a subsequence $({\mathbf{u}}_{h_j})$ of $({\mathbf{u}}_h)$, and an infinitesimal sequence $(\varepsilon _k)$ of positive levels such that ${\mathbf{u}}_{h_j}(\cdot ,\pm \varepsilon _k)$ satisfies the hypotheses (i)–(iii) of Proposition 4.9.

Let us choose also an infinitesimal sequence $(\delta _k)$ of positive numbers such that

$$\begin{aligned} \delta _k M(\varepsilon _k) \overset{\tiny {k\rightarrow +\infty }}{\longrightarrow }0 \end{aligned}$$

(4.17)

and such that $\partial _t {\mathbf{u}}(t, s)_{|s=\pm (\varepsilon _k +\delta _k)}$ exists for almost every $t\in (a,b)$ for any $k \in \mathbb N$.

Now, we define the maps ${\mathbf{v}}^k_{h_j}$ similarly to (4.9):

$$\begin{aligned} {\mathbf{v}}^k_{h_j}(t,s):= {\left\{ \begin{array}{ll} {\mathbf{u}}(t,s) &{} \mathrm{if}~ t\in (a,b),~ \varepsilon _k + \delta _k < |s|<1, \\ \frac{\varepsilon _k + \delta _k - s}{\delta _k} {\mathbf{u}}_{h_j}(t, \varepsilon _k) + \frac{s- \varepsilon _k}{\delta _k} {\mathbf{u}}(t, \varepsilon _k + \delta _k) &{} \mathrm{if}~ t\in (a,b),~ \varepsilon _k \le s \le \varepsilon _k + \delta _k, \\ \frac{\varepsilon _k + \delta _k + s}{\delta _k} {\mathbf{u}}_{h_j}(t, -\varepsilon _k) + \frac{- s - \varepsilon _k}{\delta _k} {\mathbf{u}}(t,- (\varepsilon _k + \delta _k)) &{} \mathrm{if}~ t\in (a,b),~ -(\varepsilon _k+\delta _k) \le s \le -\varepsilon _k, \\ {\mathbf{u}}_{h_j}(t,s) &{} \mathrm{if}~ t\in (a,b),~ |s|< \varepsilon _k. \end{array}\right. } \end{aligned}$$

We claim that for any $k\in \mathbb N$ we have

$$\begin{aligned} \liminf _{j\rightarrow +\infty } {\mathcal {A}}({\mathbf{v}}^k_{h_j}, \mathrm{R})\le \lim _{h \rightarrow +\infty } {\mathcal {A}}({\mathbf{u}}_h,\mathrm{R}) + C\left[ 1 + M(\varepsilon _k)\right] \delta _k, \end{aligned}$$

(4.18)

where C is the constant given in (4.10).

For any $\lambda \in (0,1)$ set $\mathrm{R}^+_\lambda := (a,b)\times (0,\lambda )$. Without loss of generality we can suppose that there exists $\displaystyle \lim _{h\rightarrow +\infty } {\mathcal {A}}({\mathbf{u}}_h, \mathrm{R}^+ {\setminus } \mathrm{R}^+_{\varepsilon _k + \delta _k})$ for any $k \in \mathbb N$. Using the same notation as in the proof of Proposition 4.9, and since ${\mathcal {A}}\left( {\mathbf{u}}_{h_j}, \mathrm{R}^+_{\varepsilon _k}\right) \le {\mathcal {A}}\left( {\mathbf{u}}_{h_j}, \mathrm{R}^+\right) - {\mathcal {A}}\left( {\mathbf{u}}_{h_j}, \mathrm{R}^+ {\setminus } \mathrm{R}^+_{\varepsilon _k + \delta _k}\right)$, we get:

$$\begin{aligned} \begin{aligned} {\mathcal {A}}\left( {\mathbf{v}}^k_{h_j}, \mathrm{R}^+\right)&= {\mathcal {A}}\left( {\mathbf{u}}, \mathrm{R}^+ {\setminus } \mathrm{R}^+_{\varepsilon _k + \delta _k}\right) + {\mathcal {A}}\left( {\mathbf{v}}^k_{h_j}, S_{\varepsilon _k}^{\varepsilon _k + \delta _k}\right) + {\mathcal {A}}\left( {\mathbf{u}}_{h_j}, \mathrm{R}^+_{\varepsilon _k}\right) \\&\le {\mathcal {A}}\left( {\mathbf{u}}, \mathrm{R}^+ {\setminus } \mathrm{R}^+_{\varepsilon _k + \delta _k}\right) - {\mathcal {A}}\left( {\mathbf{u}}_{h_j}, \mathrm{R}^+ {\setminus } \mathrm{R}^+_{\varepsilon _k + \delta _k}\right) + {\mathcal {A}}\left( {\mathbf{v}}^k_{h_j}, S_{\varepsilon _k}^{\varepsilon _k + \delta _k}\right) \\&\quad +{\mathcal {A}}\left( {\mathbf{u}}_{h_j}, \mathrm{R}^+\right) . \end{aligned} \end{aligned}$$

Passing to the limit as $j \rightarrow +\infty$, recalling that ${\mathcal {A}}(\cdot ,\mathrm{R}^+ {\setminus } \mathrm{R}^+_{\varepsilon _k + \delta _k} )$ is lower semicontinuous, using (4.10), and making similar computations also in $\mathrm{R}^-$, we get claim (4.18).

Finally, the proof of (4.2) is concluded by remembering (4.17), choosing a suitable subsequence $(k_{h_j})$ and defining ${\mathbf{v}}_{h_j}:={\mathbf{v}}_{h_j}^{k_{h_j}}$. $\square$

5 Condition II: upper bound

In this short section and in Sect. 6 we discuss the case where $\varOmega$ and ${\mathbf{u}}$ satisfy condition II. In Proposition 5.1, following the strategy of [4], we prove the upper bound in Theorem 1.4 (see inequality (5.2)). We recall that $D=[[\sigma ^-, \sigma ^+]]$ is the (fixed) domain defined in Definition 2.5. Similarly to the case when $\varOmega$ and ${\mathbf{u}}$ satisfy condition I, this upper bound implies also that $\overline{{\mathcal {A}}}_s({\mathbf{u}},\varOmega )\le m(D;\varGamma ^-, \varGamma ^+)$, since in general $\overline{{\mathcal {A}}}({\mathbf{u}},\varOmega )\le \overline{{\mathcal {A}}}^\infty ({\mathbf{u}}, \varOmega )$. In Sect. 7 we describe some examples where the latter inequality is strict.

The proof of the next proposition is similar to the one in [4, Theorem 4.1]. We briefly report it for the sake of completeness.

Proposition 5.1

(Upper bound, II) Let $\varOmega$ and ${\mathbf{u}}$ satisfy condition II. Then there exists a sequence $({\mathbf{u}}_h)\subset H^1 (\varOmega ; \mathbb {R}^2)$ converging to ${\mathbf{u}}$ in $L^1(\varOmega ;\mathbb {R}^2)$ and uniformly out of $J_{\mathbf{u}}$, such that

$$\begin{aligned} \liminf _{h \rightarrow +\infty } {\mathcal {A}}({\mathbf{u}}_h, \varOmega ) = \int _\varOmega |{\mathcal {M}}(\nabla {\mathbf{u}})|\,dt\,ds + m(D; \varGamma ^-[{\mathbf{u}}],\varGamma ^+[{\mathbf{u}}]). \end{aligned}$$

(5.1)

Hence

$$\begin{aligned} \overline{{\mathcal {A}}}^\infty _s({\mathbf{u}}, \varOmega )\le m(D;\varGamma ^-[{\mathbf{u}}],\varGamma ^+[{\mathbf{u}}]). \end{aligned}$$

(5.2)

Remark 5.2

We notice that $m(D;\varGamma ^-[{\mathbf{u}}],\varGamma ^+[{\mathbf{u}}])$ is well defined; indeed the two traces of ${\mathbf{u}}$ on the sides of $J_{\mathbf{u}}$ are defined and coincide at the endpoints of the jump, since $J_{\mathbf{u}}\subset \subset \varOmega$, and ${\mathbf{u}}\in W^{1,\infty }(\varOmega {\setminus } J_{\mathbf{u}};\mathbb {R}^2)$.

Proof

We can suppose without loss of generality that $D=[[\sigma ^-, \sigma ^+]]$ with $|\sigma ^\pm | < 1$, compare Definition 2.3, and hence $D\subset \mathrm{R}$. Let $\Phi _h \in \mathrm{semicart}(D; \varGamma ^-, \varGamma ^+)$ be such that $\lim _{h \rightarrow +\infty } \int _D |\partial _t \varPhi _h \wedge \partial _s \varPhi _h |\, dt \, ds = m(D; \varGamma ^-, \varGamma ^+)$. For any $\varepsilon \in (0,1)$ we set $\mathrm{R}_\varepsilon :=(a,b)\times (-\varepsilon ,\varepsilon )$ and $D_\varepsilon :=[[\sigma ^-_\varepsilon ,\sigma ^+_\varepsilon ]]$ with $\sigma ^\pm _\varepsilon := \varepsilon \sigma ^\pm$; we define also the map $T_\varepsilon : \mathrm{R}_\varepsilon {\setminus } D_\varepsilon \rightarrow \mathrm{R}_\varepsilon {\setminus } (a,b)\times \{0\}$ as follows:

$$\begin{aligned} T_\varepsilon (t,s):= {\left\{ \begin{array}{ll} \left( t, \frac{s-\varepsilon \sigma ^+(t)}{1-\sigma ^+(t)}\right) &{}\quad \mathrm{if}~ (t,s)\in (R_{\varepsilon } \setminus D_{\varepsilon }) \cap \{s>0\}, \\ \left( t, \frac{s-\varepsilon \sigma ^-(t)}{1+\sigma ^-(t)}\right) &{}\quad \mathrm{if}~ (t,s)\in (R_{\varepsilon } \setminus D_{\varepsilon }) \cap \{s<0\}. \end{array}\right. } \end{aligned}$$

Next, let us consider the sequence $({\mathbf{u}}^h_{\varepsilon }) \subset H^1(\varOmega ;\mathbb {R}^2)$ given by

$$\begin{aligned} {\mathbf{u}}^h_\varepsilon (t,s):= {\left\{ \begin{array}{ll} {\mathbf{u}}(t,s) &{}\quad \text{ if } (t,s)\in \Omega {\setminus } \mathrm{R}_\varepsilon \\ {\mathbf{u}}(T_\varepsilon (t,s)) &{}\quad \text{ if } (t,s) \in \mathrm{R}_\varepsilon {\setminus } \overline{D_\varepsilon } \\ \phi _h(t, s/\varepsilon ) &{}\quad \text{ if } (t,s) \in \overline{D_\varepsilon }, \end{array}\right. } \end{aligned}$$

where $\varPhi _h(t,s)=(t, \phi _h(t,s))$. Since $T_\varepsilon$ and its derivatives are bounded by a constant depending only on $\mathrm{lip}(\sigma ^\pm )$, with computations similar to the ones in Proposition 3.1 we get

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0^+}{\mathcal {A}}({\mathbf{u}}^h_\varepsilon , \varOmega ) = \int _\varOmega |{\mathcal {M}}(\nabla {\mathbf{u}})|\,\text{ d }t\,\text{ d }s + \int _D |\partial _t \varPhi _h \wedge \partial _s \varPhi _h|\, \text{ d }t \, \text{ d }s. \end{aligned}$$

Hence the required sequence is obtained as $({\mathbf{u}}_h):= ({\mathbf{u}}^h_{\varepsilon _h})$, for a suitable infinitesimal sequence $(\varepsilon _h)$ of positive numbers such that $\mathrm{R}_{\varepsilon _h}\subset \varOmega$ for every $h \in \mathbb {N}$. $\square$

6 Condition II: lower bound

In this section we want to prove inequality (6.1) which, coupled with Proposition 5.1, concludes the proof of Theorem 1.4.

Theorem 6.1

(Lower bound, II) Let $\varOmega$ and ${\mathbf{u}}$ satisfy condition II. Let $({\mathbf{u}}_h) \subset \mathrm{Lip}(\varOmega ; \mathbb {R}^2)$ be a sequence converging to ${\mathbf{u}}$ in $L^1(\varOmega ; \mathbb {R}^2)$ and uniformly out of $J_{\mathbf{u}}$. Then

$$\begin{aligned} \liminf _{h\rightarrow +\infty } {\mathcal {A}}({\mathbf{u}}_h,\varOmega ) \ge \int _\varOmega |{\mathcal {M}}(\nabla {\mathbf{u}})|\,dt\,ds + m(D;\varGamma ^-[{\mathbf{u}}],\varGamma ^+[{\mathbf{u}}]). \end{aligned}$$

Hence

$$\begin{aligned} \overline{{\mathcal {A}}}^\infty _s({\mathbf{u}},\varOmega )\ge m(D;\varGamma ^-[{\mathbf{u}}],\varGamma ^+[{\mathbf{u}}]). \end{aligned}$$

(6.1)

As in Sect. 4, we shall divide the proof in two steps: in the first step we prove the theorem under the further hypothesis that

$$\begin{aligned} {\mathbf{u}}_h={\mathbf{u}}\qquad \text{ in } \Omega {\setminus } N_h, \end{aligned}$$

(6.2)

where $(N_h)$ is a decreasing sequence of neighborhoods of $J_{\mathbf{u}}$ such that $\displaystyle \bigcap _{h \in \mathbb N} N_h=J_{\mathbf{u}}$. In order to prove this step, we shall need the analogous of Lemma 4.5. In the second step we prove that for any sequence $({\mathbf{u}}_h)$ converging to ${\mathbf{u}}$ in $L^1(\varOmega ,\mathbb {R}^2)$ and uniformly out of $J_{\mathbf{u}}$, there exists a sequence $({\mathbf{v}}_h) \subset \mathrm{Lip}(\varOmega ; \mathbb {R}^2)$ satisfying (6.2) and such that

$$\begin{aligned} \liminf _{h \rightarrow +\infty } {\mathcal {A}}({\mathbf{v}}_h,\varOmega )\le \liminf _{h \rightarrow +\infty } {\mathcal {A}}({\mathbf{u}}_h,\varOmega ). \end{aligned}$$

(6.3)

In order to prove this step we shall need the analogous of Proposition 4.9.

Let us fix some notation. For any $\varepsilon \in (0, \frac{b-a}{2})$, let $\lambda _\varepsilon : \mathbb {R}_t \rightarrow \mathbb {R}_t$ be defined as

$$\begin{aligned} \lambda _\varepsilon (t):= \frac{b-a-2\varepsilon }{b-a + 2 \varepsilon }(t-(a-\varepsilon )) + a + \varepsilon , \end{aligned}$$

so that $\lambda _\varepsilon ((a-\varepsilon , b + \varepsilon ))=(a+\varepsilon , b-\varepsilon )$. The map $\Lambda _\varepsilon : \mathbb {R}^2_{(t,s)} \rightarrow \mathbb {R}^2_{(t,s)}$ is, instead, defined as

$$\begin{aligned} \Lambda _\varepsilon (t,s)= (\lambda _\varepsilon (t),s). \end{aligned}$$

We set $O_\varepsilon :=[[\sigma ^-_\varepsilon , \sigma ^+_\varepsilon ]]$, where $\sigma ^\pm _\varepsilon \in \mathrm{Lip}([a-\varepsilon , b+\varepsilon ])$ are such that $\sigma ^-_\varepsilon (a-\varepsilon )= \sigma ^+_\varepsilon (a-\varepsilon )$ and $\sigma ^-_\varepsilon (b+\varepsilon )= \sigma ^+_\varepsilon (b+\varepsilon )$, and such that $\Lambda _\varepsilon (O_\varepsilon )\subset \subset D$, see Fig. 4. If necessary, we can require $\partial O_\varepsilon$ without horizontal cusps, in the sense that we can suppose that there is a decreasing sequence $(t_i)\subset (a,b)$ converging to a, along which $\sigma ^\pm _\varepsilon$ are differentiable, and $\liminf _{i \rightarrow + \infty }(\sigma ^+_\varepsilon )'(t_i)> 0$, $\limsup _{i \rightarrow + \infty }(\sigma ^-_\varepsilon )'(t_i) < 0$ (and similarly near $t=b$).

Fig. 4

For any $\varepsilon >0$ small enough, $O_\varepsilon =[[\sigma ^-_\varepsilon ,\sigma ^+_\varepsilon ]]$ is such that its image through the map $\Lambda _\varepsilon$ is compactly contained in the fixed domain D

Lemma 6.2

Let $\gamma ^\pm \in \mathrm{Lip}([a,b];\mathbb {R}^2)$ be such that $\gamma ^-(a)=\gamma ^+(a)$ and $\gamma ^-(b)=\gamma ^+(b)$. Let $(\varepsilon _h)$ be an infinitesimal sequence of positive numbers and let $\gamma ^\pm _{h}\in \mathrm{Lip}([a-\varepsilon _h, b+ \varepsilon _h];\mathbb {R}^2)$ be maps with the following properties:

• (i) $\gamma ^-_h(a-\varepsilon _h)=\gamma ^+_h(a-\varepsilon _h)$ and $\gamma ^-_h(b+\varepsilon _h)=\gamma ^+_h(b+\varepsilon _h)$ for any $h\in \mathbb {N}$;

• (ii) $\displaystyle \lim _{h \rightarrow +\infty } \gamma _h^-(a-\varepsilon _h)= \gamma ^-(a)$ and $\displaystyle \lim _{h \rightarrow +\infty } \gamma ^-_h(b+\varepsilon _h)= \gamma ^-(b)$;

• (iii) $\displaystyle \lim _{h \rightarrow +\infty }||\gamma ^\pm _{h}\circ \lambda ^{-1}_{\varepsilon _h} - \gamma ^\pm ||_{L^1((a+\varepsilon _h,b-\varepsilon _h);\mathbb {R}^2)} = 0$.

Moreover, we also suppose:

• (iv) there exists a constant $C_1>0$ such that $||\dot{\gamma }^\pm _{_h}||_{L^\infty ((a-\varepsilon _h,b+\varepsilon _h);\mathbb {R}^2)} \le C_1$ for any $h \in \mathbb N$.

Then

$$\begin{aligned} m(D;\varGamma ^-, \varGamma ^+)\le \liminf _{h\rightarrow +\infty } m(O_{\varepsilon _h};\varGamma ^-_{h}, \varGamma ^+_{h}), \end{aligned}$$

(6.4)

where $\varGamma ^\pm := \mathrm{graph}(\gamma ^\pm )$, $\varGamma ^\pm _{h}:= \mathrm{graph}(\gamma ^\pm _{h})$.⁸

Proof

Let $\varPsi _{h}$ be a semicartesian map in $H^1(O_{\varepsilon _h};\mathbb {R}^3)$ spanning $\varGamma _{h}$ such that

$$\begin{aligned} \int _{O_{\varepsilon _h}}|\partial _t \varPsi _{h} \wedge \partial _s \varPsi _{h}|\, \text{ d }t \, \text{ d }s \le m(O_{\varepsilon _h}; \varGamma ^-_{h}, \varGamma ^+_{h}) + \varepsilon _h, \end{aligned}$$

(6.5)

with $\varPsi _h(t,s)=(t,\psi _h(t,s))$. Let us define $\varPhi _h\in H^1(\Lambda _{\varepsilon _h}(O_{\varepsilon _h});\mathbb {R}^3)$ as

$$\begin{aligned} \varPhi _h(t,s) := (t, \psi _h(\lambda ^{-1}_{\varepsilon _h}(t),s))=: (t, \phi _h(t,s)), \qquad (t,s) \in \Lambda _{\varepsilon _h}(O_{\varepsilon _h}). \end{aligned}$$

In words, we start from a point in $\Lambda _{\varepsilon _h}(O_{\varepsilon _h})$, we take its image in $O_{\varepsilon _h}$ through the dilation $\Lambda ^{-1}_{\varepsilon _h}$, we pass to its image through the semicartesian map $\varPsi _h$, and we contract in the t-direction through the map $(t,\xi ,\eta )\rightarrow (\lambda _{\varepsilon _h}(t), \xi ,\eta )$. Recalling (6.5) and since the determinant of the Jacobian of $\Lambda _{\varepsilon _h}$ tends to 1 as $h\rightarrow +\infty$, we get

$$\begin{aligned} \int _{\Lambda _{\varepsilon _h}(O_{\varepsilon _h})}|\partial _t \varPhi _h \wedge \partial _s \varPhi _h|\,\text{ d }t \,\text{ d }s = m(O_{\varepsilon _h}; \varGamma ^-_h,\varGamma ^+_h) + \mathcal {O}(\varepsilon _h). \end{aligned}$$

(6.6)

Recalling that $\Lambda _{\varepsilon _h}(O_{\varepsilon _h})\subset \subset D$, we can extend $\varPhi _h$ to a semicartesian map in $\mathrm{semicart}(D;\varGamma ^-,\varGamma ^+)$: if $\Lambda (O_{\varepsilon _h}):=[[\sigma ^-_{\varepsilon _h}, \sigma ^+_{\varepsilon _h}]]$, we define $\varPhi _h$ in $S^+_{\varepsilon _h}:=\{(t,s)\in D:\, t\in (a+\varepsilon _h, b-\varepsilon _h),\, s\in (\sigma ^+_{\varepsilon _h}(t), \sigma ^+(t))\}$ as

$$\begin{aligned} \varPhi _h(t,s):= \left( t, \frac{s - \sigma ^+_{\varepsilon _h}(t)}{\sigma ^+(t)- \sigma ^+_{\varepsilon _h}(t)}\gamma ^+(t) + \frac{\sigma ^+(t)- s}{\sigma ^+(t)-\sigma ^+_{\varepsilon _h}(t)} \phi _h(t, \sigma ^+_{\varepsilon _h}(t)) \right) . \end{aligned}$$

Similarly, we define $\varPhi _h$ on $S^-_{\varepsilon _h}:=\{(t,s)\in D:\, t\in (a+\varepsilon _h, b-\varepsilon _h),\, s\in (\sigma ^-(t), \sigma ^-_{\varepsilon _h}(t))\}$. Thanks to hypotheses and recalling Lemma 4.4 (see inequality (4.4)), we deduce

$$\begin{aligned} \int _{S^-_{\varepsilon _h} \cup S^+_{\varepsilon _h}}|\partial _t \varPhi _h \wedge \partial _s \varPhi _h|\,\text{ d }t\,\text{ d }s \overset{h\rightarrow +\infty }{\longrightarrow }0. \end{aligned}$$

(6.7)

Now, we define $\varPhi _h$ on the curved triangles $T^a_h:=\{(t,s)\in D:\, t\in (a,a+\varepsilon _h]\}$ and $T^b_h:=\{(t,s)\in D:\, t\in [b-\varepsilon _h,b)\}$. Let us define $f_h^a \in \mathrm{Lip}([a,a+\varepsilon _h];\mathbb {R}^2)$ as

$$\begin{aligned} f_h^a(t):= \frac{\phi _h(a+\varepsilon _h, 0) - \gamma ^+(a)}{\varepsilon _h}(t-a) + \gamma ^+(a), \end{aligned}$$

so that its graph is the segment joining $(a,\gamma ^+(a))$ and $(a+\varepsilon _h, \phi _h(a+\varepsilon _h,0))$. Next, for $(t,s)\in T^a_h$, set

$$\begin{aligned} \varPhi _h(t,s):= {\left\{ \begin{array}{ll} \left( t, \frac{s}{\sigma ^+(t)}\gamma ^+(t) + \frac{\sigma ^+(t) - s}{\sigma ^+(t)}f_h^a(t) \right) \,\, &{}\quad \mathrm{if}~ s\ge 0, \\ \left( t, \frac{s}{\sigma ^-(t)}\gamma ^-(t) + \frac{\sigma ^-(t) - s}{\sigma ^-(t)}f_h^a(t)\right) \,\, &{}\quad \mathrm{if}~ s < 0, \end{array}\right. } \end{aligned}$$

and similarly on $T^b_h$. Again, Lemma 4.4 and our hypotheses imply

$$\begin{aligned} \int _{T^a_h \cup T^b_h}|\partial _t \varPhi _h \wedge \partial _s \varPhi _h| \,\text{ d }t\,\text{ d }s \overset{h\rightarrow +\infty }{\longrightarrow }0. \end{aligned}$$

(6.8)

Thus, using (6.6), (6.7) and (6.8) we obtain, for any $h \in \mathbb N$,

$$\begin{aligned} m(D;\varGamma ^-,\varGamma ^+)\le \int _{D}|\partial _t \varPhi _h \wedge \partial _s \varPhi _h |\,\text{ d }t \, \text{ d }s\le m(O_{\varepsilon _h};\varGamma ^-_h,\varGamma ^+_h) + \mathcal {O}(\varepsilon _h). \end{aligned}$$

Passing to the limit as $h \rightarrow +\infty$, (6.4) follows. $\square$

For any $d>0$, define $J_{\mathbf{u}}^d:=\{(t,s)\in \mathbb {R}^2: \, \mathrm{dist}((t,s), J_{\mathbf{u}})< d\}$. We parametrize the curve $\{s>0\} \cap \partial J_{\mathbf{u}}^d$ on the interval $(a-\frac{\pi }{2}, b+\frac{\pi }{2})$ by the map $\beta ^+_d$ defined by

$$\begin{aligned} \beta ^+_d(\theta ):= {\left\{ \begin{array}{ll} (a+d \sin (\theta -a), d\cos (\theta -a)) &{}\quad \text{ if }\quad \theta \in (a-\pi /2,a), \\ (\theta , d) &{}\quad \text{ if }\quad \theta \in [a,b], \\ (b+d \sin (\theta - b), d\cos (\theta -b)) &{}\quad \text{ if }\quad \theta \in (b,b+\pi /2), \end{array}\right. } \end{aligned}$$

and, similarly, we define the parametrization $\beta ^-_d$ for $\{s < 0\} \cap \partial J_{\mathbf{u}}^d$. We can now introduce the coordinates $(\theta , r)$ in $\mathbb {R}^2{\setminus } \{s=0\}$ such that $(t,s)=\beta _{r}^+(\theta )$   if $s>0$, and $(t,s)=\beta _{r}^-(\theta )$   if $s<0$.

Proposition 6.3

(Interpolation, II) Let $\varOmega ^+:=\varOmega \cap \{s>0\}$, let $({\mathbf{u}}_h)\subset \mathrm{Lip}(\varOmega ^+; \mathbb {R}^2)$, ${\mathbf{u}}\in \mathrm{Lip}(\varOmega ^+; \mathbb {R}^2)$, and suppose that ${\mathbf{u}}_h \rightarrow {\mathbf{u}}$ in $L^1(\varOmega ^+; \mathbb {R}^2)$ as $h \rightarrow +\infty$. Let $\varepsilon _o>0$ be fixed so that $\overline{J_{\mathbf{u}}^{\varepsilon _o}} \cap \{s>0\} \subset \varOmega ^+$. For any $\varepsilon \in (0,\varepsilon _o]$ we define $\gamma ^\varepsilon _h:= {\mathbf{u}}_h \circ \beta ^+_\varepsilon$ and $\gamma ^\varepsilon := {\mathbf{u}}\circ \beta ^+_\varepsilon$. Let us suppose that $\dot{\gamma }^{\varepsilon _o}$ exists almost everywhere in $I:=(a-\frac{\pi }{2}, b+\frac{\pi }{2})$, and let $\varepsilon _i\in (0,\varepsilon _o)$ be such that:

• (i) $||\gamma ^{\varepsilon _i}_{h} - \gamma ^{\varepsilon _i}||_{L^\infty (I;\mathbb {R}^2)} \rightarrow 0$ as $h\rightarrow +\infty$;

• (ii) $\dot{\gamma }^{\varepsilon _i}_{h}$ exists almost everywhere in I for any $h\in \mathbb {N}$;

• (iii) $\displaystyle \liminf _{h\rightarrow +\infty }||\dot{\gamma }^{\varepsilon _i}_{h}||_{L^1(I;\mathbb {R}^2)}\le M$, where the constant M may depend on $\varepsilon _i$.

Let us define the sequence $({\mathbf{v}}_h)\subset \mathrm{Lip}(\varOmega ^+;\mathbb {R}^2)$ as ${\mathbf{v}}_h:={\mathbf{u}}$ on $\varOmega ^+{\setminus } J_{\mathbf{u}}^{\varepsilon _o}$, ${\mathbf{v}}_h:={\mathbf{u}}_h$ in $\varOmega ^+\cap J_{\mathbf{u}}^{\varepsilon _i}$, and such that its representation in $(\theta ,r)$ coordinates in the curvilinear strip $S_{\varepsilon _i}^{\varepsilon _o}:= \varOmega ^+ \cap \left( J_{\mathbf{u}}^{\varepsilon _o}{\setminus }J_{\mathbf{u}}^{\varepsilon _i}\right)$ is

$$\begin{aligned} \tilde{{\mathbf{v}}}_h(\theta ,r):=\frac{\varepsilon _o - r}{\varepsilon _o - \varepsilon _i} \gamma ^{\varepsilon _i}_{h}(\theta ) + \frac{r-\varepsilon _i}{\varepsilon _o-\varepsilon _i}\gamma ^{\varepsilon _o}(\theta ). \end{aligned}$$

Then

$$\begin{aligned} \liminf _{h\rightarrow +\infty } {\mathcal {A}}\Big ({\mathbf{v}}_h, \varOmega ^+ \cap \left( J_{\mathbf{u}}^{\varepsilon _o}{\setminus }J_{\mathbf{u}}^{\varepsilon _i}\right) \Big )\le C(1 + M)|\varepsilon _o - \varepsilon _i|, \end{aligned}$$

(6.9)

where $C=C(\mathrm{lip}({\mathbf{u}}))$.

Proof

The term ${\mathcal {A}}({\mathbf{v}}_h, S_{\varepsilon _i}^{\varepsilon _o}\cap \{t\in (a,b)\})$ can be estimated by the right hand side of (6.9) using Proposition 4.9, since in $S_{\varepsilon _i}^{\varepsilon _o}\cap \{t\in (a,b)\}$ we have $\theta (t,s)=t$ and $r(t,s)=s$.

We prove the estimate for ${\mathcal {A}}({\mathbf{v}}_h, S_{\varepsilon _i}^{\varepsilon _o}\cap \{t<a\})$, the computations for ${\mathcal {A}}({\mathbf{v}}_h, S_{\varepsilon _i}^{\varepsilon _o}\cap \{t>b\})$ being similar. We have:

$$\begin{aligned} \begin{aligned} {\mathcal {A}}({\mathbf{v}}_{h}, S_{\varepsilon _i}^{\varepsilon _o}\cap \{t< a \})&= \int _{\varepsilon _i}^{\varepsilon _o}\int _{a-\pi /2}^a \sqrt{r^2 + |\partial _\theta \tilde{{\mathbf{v}}}_{h}|^2 + r^2 |\partial _r \tilde{{\mathbf{v}}}_{h}|^2+ (\mathrm{det} \nabla _{\theta , r}\tilde{{\mathbf{v}}}_{h})^2}~\hbox {d}\theta \,\hbox {d}r \\&\le C \int _{\varepsilon _i}^{\varepsilon _o} \int _{a-\pi /2}^a \big [ r + |\partial _\theta \tilde{{\mathbf{v}}}_h| + r|\partial _r \tilde{{\mathbf{v}}}_h| + |\mathrm{det} \nabla _{\theta , r}\tilde{{\mathbf{v}}}_{h}| \big ]\, \hbox {d}\theta \,\hbox {d}r \end{aligned} \end{aligned}$$

where $\nabla _{\theta , r}$ denotes the Jacobian with respect to $(\theta ,r)$, and C is an absolute positive constant. Again we estimate the right hand side as in the proof of Proposition 4.9 and using our assumptions:

$$\begin{aligned}&-\int _{\varepsilon _i}^{\varepsilon _o}\int _{a-\pi /2}^{\pi }r\,\hbox {d}r\,\hbox {d}\theta = \pi /2(\varepsilon _o^2-\varepsilon _i^2),\\&-\int _{\varepsilon _i}^{\varepsilon _o}\int _{a-\pi /2}^a|\partial _\theta \tilde{{\mathbf{v}}}_{h}|\, \hbox {d}\theta \,\hbox {d}r \le \int _{\varepsilon _i}^{\varepsilon _o}\int _{a-\pi /2}^a\left[ |\dot{\gamma }^{\varepsilon _o}| + |\dot{\gamma }^{\varepsilon _i}_{h}|\right] \, \hbox {d}\theta \,\hbox {d}r \le \pi \varepsilon _o \mathrm{lip}({\mathbf{u}})(\varepsilon _o -\varepsilon _i)+ (\varepsilon _o - \varepsilon _i)M,\\&-\int _{\varepsilon _i}^{\varepsilon _o}\int _{a-\pi /2}^a r |\partial _r \tilde{{\mathbf{v}}}_{h}|\,\hbox {d}\theta \,\hbox {d}r = \int _{\varepsilon _i}^{\varepsilon _o}\int _{a-\pi /2}^a r \frac{|\gamma ^{\varepsilon _o}-\gamma ^{\varepsilon _i}_{h}|}{\varepsilon _o-\varepsilon _i}\,\hbox {d}\theta \,\hbox {d}r\\&\le \int _{\varepsilon _i}^{\varepsilon _o}\int _{a-\pi /2}^a r \frac{|\gamma ^{\varepsilon _o}-\gamma ^{\varepsilon _i}|+|\gamma ^{\varepsilon _i} - \gamma ^{\varepsilon _i}_{h}|}{\varepsilon _o-\varepsilon _i}\,\hbox {d}\theta \,\hbox {d}r \le \frac{\pi }{4} \mathrm{lip}({\mathbf{u}}) (\varepsilon _o^2 - \varepsilon _i^2)\\&+ \frac{\varepsilon _o+ \varepsilon _i}{2}\int _{a-\pi /2}^a |\gamma ^{\varepsilon _i} - \gamma ^{\varepsilon _i}_{h}|\,\hbox {d}\theta ,\\&- \int _{\varepsilon _i}^{\varepsilon _o} \int _{a-\pi /2}^a |\mathrm{det} \nabla _{\theta ,r} \tilde{{\mathbf{v}}}_{h}|\,\hbox {d}\theta \,\hbox {d}r \le 2\int _{\varepsilon _i}^{\varepsilon _o} \int _{a-\pi /2}^a |\partial _r \tilde{{\mathbf{v}}}_{h}||\partial _\theta \tilde{{\mathbf{v}}}_{h}|\,\hbox {d}\theta \, \hbox {d}r \le 2\int _{a-\pi /2}^a |\gamma ^{\varepsilon _o} - \gamma ^{\varepsilon _i}_{h}||\dot{\gamma }^{\varepsilon _o}+ \dot{\gamma }^{\varepsilon _i}_{h}| \, \hbox {d}\theta \\&\le \int _{a-\pi /2}^a |\dot{\gamma }^{\varepsilon _o}||\gamma ^{\varepsilon _o} - \gamma ^{\varepsilon _i}_{h}|\,\hbox {d} \theta + \int _{a-\pi /2}^a |\dot{\gamma }^{\varepsilon _i}_{h}||\gamma ^{\varepsilon _o} - \gamma ^{\varepsilon _i}_{h}|\,\hbox {d} \theta \\&\le \varepsilon _o \mathrm{lip}({\mathbf{u}}) \big (\mathrm{lip}({\mathbf{u}})|\varepsilon _o - \varepsilon _i| + ||\gamma ^{\varepsilon _i}-\gamma ^{\varepsilon _i}_{h}||_{L^1(I,\mathbb {R}^2)}\big ) + M \left( \mathrm{lip}({\mathbf{u}})|\varepsilon _o - \varepsilon _i|+ ||\gamma ^{\varepsilon _i}- \gamma ^{\varepsilon _i}_{h}||_{L^{\infty }(I;\mathbb {R}^2)} \right) . \end{aligned}$$

Using our assumptions and the previous estimates, we get

$$\begin{aligned} \liminf _{h \rightarrow +\infty } {\mathcal {A}}({\mathbf{v}}_{h}, S_{\varepsilon _i}^{\varepsilon _o}\cap \{t<a\})\le C(1 + M)|\varepsilon _o - \varepsilon _i|, \end{aligned}$$

where $C=C(\mathrm{lip}({\mathbf{u}}))$. $\square$

We are now in the position to prove Theorem 6.1.

Proof

Let us suppose first that $({\mathbf{u}}_h)$ satisfies (6.2). Let $(\varepsilon _h)$ and $(\tilde{\varepsilon }_h)$ be two infinitesimal sequences of positive numbers such that

$$\begin{aligned} N_h \subset \subset \left\{ (t,s): \left( t, \frac{s}{\tilde{\varepsilon }_h}\right) \in O_{\varepsilon _h}\right\} \subset \varOmega , \end{aligned}$$

where $O_{\varepsilon _h}=[[\sigma ^-_{\varepsilon _h}, \sigma ^+_{\varepsilon _h}]]$ is defined as in Lemma 6.2. Let $\gamma ^\pm _h \in \mathrm{Lip}([a-\varepsilon _h, b+\varepsilon _h];\mathbb {R}^2)$ be defined as

$$\begin{aligned} \gamma ^\pm _h(t):={\mathbf{u}}_h\left( t, \tilde{\varepsilon }_h\sigma ^\pm _{\varepsilon _h}(t)\right) ={\mathbf{u}}\left( t, \tilde{\varepsilon }_h\sigma ^\pm _{\varepsilon _h}(t)\right) . \end{aligned}$$

Following the same computation as in Lemma 4.6 we get

$$\begin{aligned} {\mathcal {A}}\left( {\mathbf{u}}_h, \left\{ (t,s): \left( t, \frac{s}{\tilde{\varepsilon }_h}\right) \in O_{\varepsilon _h}\right\} \right) \ge m(O_{\varepsilon _h};\varGamma ^-_h,\varGamma ^+_h), \end{aligned}$$

where $\varGamma ^\pm _h:=\mathrm{graph}(\gamma ^\pm _h)$. Due to the regularity assumptions on ${\mathbf{u}}$, the sequences $(\gamma ^\pm _h)$ satisfy the hypotheses of Lemma 6.2, and thus we can conclude that

$$\begin{aligned} \liminf _{h\rightarrow +\infty } {\mathcal {A}}({\mathbf{u}}_h,\varOmega )&\ge \int _\varOmega |{\mathcal {M}}(\nabla {\mathbf{u}})|\,\text{ d }t\,\text{ d }s + \liminf _{h\rightarrow +\infty }m(O_{\varepsilon _h};\varGamma ^-_h,\varGamma ^+_h)\\&\ge \int _\varOmega |{\mathcal {M}}(\nabla {\mathbf{u}})|\,\text{ d }t\,\text{ d }s + m(D; \varGamma ^-,\varGamma ^+). \end{aligned}$$

Now, we have to prove that for any sequence $({\mathbf{u}}_h)$ converging to ${\mathbf{u}}$ in $L^1(\varOmega ;\mathbb {R}^2)$ and uniformly out of $J_{\mathbf{u}}$ we can build a sequence $({\mathbf{v}}_h)$ satisfying (6.2) and (6.3). The proof follows along the same lines of Theorem 4.3, where the choice of an infinitesimal sequence $(\varepsilon _k) \subset (0,+\infty )$ satisfying hypotheses (i)–(iii) of Proposition 6.3 is guaranteed by Fatou’s lemma applied to the area functional in the $(\theta ,r)$ coordinates. $\square$

7 Examples for which $\overline{{\mathcal {A}}}<\overline{{\mathcal {A}}}^\infty$

In this section we exhibit some examples of pairs $(\varOmega , {\mathbf{u}})$ satisfying condition II and for which $\overline{{\mathcal {A}}}({\mathbf{u}},\varOmega )< \overline{{\mathcal {A}}}^\infty ({\mathbf{u}}, \varOmega )$. The idea is that, under certain circumstances, sequences converging to ${\mathbf{u}}$ in $L^1(\varOmega ;\mathbb {R}^2)$, but not uniformly out of $J_{\mathbf{u}}$, can provide an upper bound lower than the right hand side of (5.1). What is suggested by these examples is that we could extend in some way the jump, adding to $J_{\mathbf{u}}$ a sort of “virtual” jump, and build sequences converging uniformly to ${\mathbf{u}}$ out of this extension. How choosing these extensions seems not easy. We present different possibilities that confirm the strong non-local behavior of the functional $\overline{{\mathcal {A}}}({\mathbf{u}}, \cdot )$.

7.1 Virtual jump starting from an endpoint of $J_{\mathbf{u}}$

In [1, Section 5] the authors study $\overline{{\mathcal {A}}}({{\mathbf{u}}_V},B_R)$, with ${{\mathbf{u}}_V}(t,s):= \frac{(t,s)}{|(t,s)|}$, $(t,s)\in \mathbb {R}^2{\setminus } \{0\}$ (the vortex map), providing two different upper bounds. In [1, Lemma 5.2] they bound $\overline{{\mathcal {A}}}_s({{\mathbf{u}}_V},B_R)$ by the measure of the 2-dimensional unit disk, while in [1, Lemma 5.3] by the lateral area of a cylinder, whose height is the distance between the vortex and the boundary of $B_R$ (namely, R), and whose basis is the unit disk.

The idea of [1, Lemma 5.3] is the following. Let us express the map ${{\mathbf{u}}_V}$ in polar coordinates $(r,\theta )$, without renaming it, i.e., ${{\mathbf{u}}_V}(r, \theta ) = (\cos \theta , \sin \theta )$. Let $(\theta _h)$ and $(r_h)$ be two infinitesimal sequences of positive numbers, and let us define the functions $f_h : [-\pi , \pi ] \rightarrow \mathbb {R}$, $g_h : [0,R) \rightarrow [0,+\infty )$ as

$$\begin{aligned} f_h(\theta ):= {\left\{ \begin{array}{ll} -\frac{\pi -\theta _h}{\theta _h} (\theta +\pi ) &{} \text{ if } \theta \in [-\pi , -\pi + \theta _h), \\ \theta \ &{} \text{ if } \theta \in [-\pi +\theta _h,\pi - \theta _h),\\ -\frac{\pi -\theta _h}{\theta _h} (\theta -\pi ) &{} \text{ if } \theta \in [\pi - \theta _h, \pi ), \end{array}\right. } \qquad g_h(r):= {\left\{ \begin{array}{ll} \frac{r}{r_h} &{} \text{ if } r\in (0,r_h], \\ 1 &{} \text{ if } r\in (r_h, R). \end{array}\right. } \end{aligned}$$

Then, the sequence $({\mathbf{u}}_h)\subset \mathrm{Lip}(B_R;\mathbb {R}^2)$ defined by

$$\begin{aligned} {\mathbf{u}}_h(r,\theta ):=\Big (\cos \big (g_h(r)f_h(\theta )\big ), \sin \big (g_h(r)f_h(\theta )\big ) \Big ) \end{aligned}$$

converges to ${{\mathbf{u}}_V}$ in $L^1(B_R;\mathbb {R}^2)$ and $\lim _{h\rightarrow +\infty } {\mathcal {A}}({\mathbf{u}}_h, B_R) \le \int _{B_R}|{\mathcal {M}}(\nabla {{\mathbf{u}}_V})|\, \text{ d }t\,\text{ d }s + 2 \pi R$.⁹

We adapt the procedure of [1, Lemma 5.3] when $\varOmega$ and ${\mathbf{u}}$ satisfy condition II, and we build a sequence $({\mathbf{u}}_h)\subset \mathrm{Lip}(\mathrm{R};\mathbb {R}^2)$ converging to ${\mathbf{u}}$ in $L^1(\varOmega ;\mathbb {R}^2)$ and uniformly out of a curve containing $J_{\mathbf{u}}$ and having an endpoint on $\partial \varOmega$. In this case the virtual jump connects one endpoint of $J_{\mathbf{u}}$ and $\partial \varOmega$. The singular contribution

$$\begin{aligned} \lim _{h\rightarrow +\infty } {\mathcal {A}}({\mathbf{u}}_h,\varOmega ) - \int _\varOmega |{\mathcal {M}}(\nabla {\mathbf{u}})|\,\hbox {d}t\,\hbox {d}s \end{aligned}$$

can be interpreted as the area of a suitable semicartesian parametrization (Remark 7.2) with non-empty partially free boundary and, under certain circumstances, it is lower than $m(D;\varGamma ^-[{\mathbf{u}}], \varGamma ^+[{\mathbf{u}}])$, see inequality (7.10); compare also with (1.6).

Proposition 7.1

Let $\varOmega$ and ${\mathbf{u}}$ satisfy condition II, with the further conditions that $\varOmega \cap \{s=0\}= (a_1, b + \delta )\times \{0\}$ for some $a_1<a$ and $\delta >0$, and

$$\begin{aligned} {\mathbf{u}}\in {\mathcal {C}}^1\left( \overline{ \varOmega \cap \{s>0\}};\mathbb {R}^2\right) \cap {\mathcal {C}}^1\left( \overline{ \varOmega \cap \{s<0\}};\mathbb {R}^2\right) . \end{aligned}$$

(7.1)

Then there exists a sequence $({\mathbf{u}}_h)\subset \mathrm{Lip}(\varOmega ;\mathbb {R}^2)$ converging to ${\mathbf{u}}$ in $L^1(\varOmega ;\mathbb {R}^2)$ such that

$$\begin{aligned}&\lim _{h\rightarrow +\infty } {\mathcal {A}}({\mathbf{u}}_h,\varOmega ) \le \int _\varOmega |{\mathcal {M}}(\nabla {\mathbf{u}})|\,\hbox {d}t\,\hbox {d}s\nonumber \\&\quad + (b+\delta -a)\left\{ \int _a^b \Big [|\dot{\gamma }^-[{\mathbf{u}}]|+|\dot{\gamma }^+[{\mathbf{u}}]|\Big ]\,\hbox {d}t + 2 \int _b^{b+\delta }|\partial _t {\mathbf{u}}(t,0)|\,\hbox {d}t \right\} . \end{aligned}$$

(7.2)

Hence

$$\begin{aligned} \overline{{\mathcal {A}}}_s({\mathbf{u}},\Omega ) \le (b+\delta -a)\left\{ \int _a^b \Big [|\dot{\gamma }^-[{\mathbf{u}}]|+ |\dot{\gamma }^+[{\mathbf{u}}]|\Big ]\,\hbox {d}t + 2 \int _b^{b+\delta }|\partial _t {\mathbf{u}}(t,0)|\,\hbox {d}t \right\} . \end{aligned}$$

(7.3)

Proof

Given an infinitesimal sequence $(\varepsilon _h)$ of positive numbers, define (Fig. 5)

$$\begin{aligned} C_{\varepsilon _h}:=\{(t,s)\in \varOmega : \, t>a, \, |s| < \varepsilon _h (t-a)\}, \end{aligned}$$

(7.4)

and $r_{\varepsilon _h}:\varOmega \rightarrow \varOmega {\setminus } C_{\varepsilon _h}$ as

$$\begin{aligned} r_{\varepsilon _h}(t,s):= {\left\{ \begin{array}{ll} (t,s)\, &{} (t,s) \in \varOmega {\setminus } C_{\varepsilon _h}, \\ \left( \frac{s}{\varepsilon _h} +a ,s \right) \, &{} (t,s)\in C_{\varepsilon _h},\,s\ge 0, \\ \left( - \frac{s}{\varepsilon _h} +a ,s \right) \, &{} (t,s)\in C_{\varepsilon _h},\,s < 0, \end{array}\right. } \end{aligned}$$

(7.5)

that is the retraction mapping each point $(t,s) \in C_{\varepsilon _h}$ into the point of $\partial C_{\varepsilon _h} \cap \varOmega$ having s as second coordinate.

Let us define the sequence $({\mathbf{u}}_h)\subset \mathrm{Lip}(\varOmega ;\mathbb {R}^2)$ as

$$\begin{aligned} {\mathbf{u}}_h(t,s):= {\mathbf{u}}(r_{\varepsilon _h}(t,s)), \qquad (t,s) \in \Omega . \end{aligned}$$

(7.6)

We observe that $({\mathbf{u}}_h)$ converges to ${\mathbf{u}}$ in $L^1(\varOmega ;\mathbb {R}^2)$ but not uniformly out of $J_{\mathbf{u}}$, as $h \rightarrow +\infty$. Incidentally, we notice that $({\mathbf{u}}_h)$ converges to ${\mathbf{u}}$ uniformly out of a suitable “extension” of the jump, $J_\mathrm{ext}:=(a,b+\delta )\times \{0\}$.

Fig. 5

The set $\varOmega$ and, in gray, the triangle $C_{\varepsilon _h}$ built in Proposition 7.1. The map ${\mathbf{u}}_h$ defined in (7.6) is constant on the horizontal segments in $C_{\varepsilon _h}$. The sequence $({\mathbf{u}}_h)$ converges to ${\mathbf{u}}$ in $L^1(\varOmega ;\mathbb {R}^2)$ and uniformly out of the segment $J_\mathrm{ext}=(a,b+\delta )\times \{0\}$, union of $J_{\mathbf{u}}$ (the bold segment) and of the virtual jump $[b,b+\delta ]\times \{0\}$, represented by a bold dotted line

Denoting by $\partial _1 {\mathbf{u}}$ and $\partial _2 {\mathbf{u}}$ the derivative with respect to the first and second variable of ${\mathbf{u}}$, let us compute the area of the graph of ${\mathbf{u}}_{h}$ on $C_{\varepsilon _h}\cap \{s\ge 0\}$ ¹⁰:

$$\begin{aligned} \begin{aligned}&{\mathcal {A}}({\mathbf{u}}_h, C_{\varepsilon _h}\cap \{s\ge 0\})\\&\quad = \int _{a}^{b+\delta } \int _0^{\varepsilon _h(t-a)} \sqrt{1 +\frac{1}{\varepsilon _h^2}\left| \partial _1 {\mathbf{u}}\left( \frac{s}{\varepsilon _h}+a , s\right) \right| ^2 + \left| \partial _2 {\mathbf{u}}\left( \frac{s}{\varepsilon _h}+a , s\right) \right| ^2}~\hbox {d}s \, \hbox {d}t \\&\quad =\int _a^{b+\delta } \int _0^{\varepsilon _h(t-a)} \frac{1}{\varepsilon _h}\sqrt{\left| \partial _1 {\mathbf{u}}\left( \frac{s}{\varepsilon _h}+a , s\right) \right| ^2 + \mathcal {O}(\varepsilon _h^{2})} ~\hbox {d}s\,\hbox {d}t \\&\quad = \int _a^{b+\delta }\int _a^t \sqrt{| \partial _1 {\mathbf{u}}(\tau , \varepsilon _h(\tau - a))|^2 + \mathcal {O}(\varepsilon _h^{2})} ~\hbox {d}\tau \,\hbox {d}t \\&\quad \le (b+\delta - a )\int _a^{b+\delta } \sqrt{| \partial _1 {\mathbf{u}}(\tau , \varepsilon _h(\tau - a))|^2 + \mathcal {O}(\varepsilon _h^{2})} ~\hbox {d}\tau . \end{aligned} \end{aligned}$$

Similarly

$$\begin{aligned} {\mathcal {A}}({\mathbf{u}}_h, C_{\varepsilon _h}\cap \{s< 0\}) \le (b+\delta - a )\int _a^{b+\delta } \sqrt{| \partial _1 {\mathbf{u}}(\tau , -\varepsilon _h(\tau - a))|^2 + \mathcal {O}(\varepsilon _h^{2})} ~\hbox {d}\tau . \end{aligned}$$

Thus, noticing that $|\partial _1 {\mathbf{u}}(\tau , \pm \varepsilon _h(\tau - a))|$ are uniformly bounded and that $\partial _1 {\mathbf{u}}(\tau , \pm \varepsilon _h(\tau - a)) \rightarrow \dot{\gamma }^\pm [{\mathbf{u}}](\tau )$ pointwise for $\tau \in (a,b)$, and $\partial _1 {\mathbf{u}}(\tau , \pm \varepsilon _h(\tau - a))= \partial _1{\mathbf{u}}(\tau ,0)$ in $(b, b+\delta )$, possibly passing to a subsequence we get (7.2) $\square$

Remark 7.2

(Semicartesian interpretation)The right hand side of (7.3) can be interpreted as the area of the semicartesian parametrization built in Example 2.12, with $b+\delta$ in place of b and with $\mathrm{C}:= \gamma ([0, 2(b+\delta - a)])$, where $\gamma :[0, 2(b+\delta - a)] \rightarrow \mathbb {R}^2$ is defined as follows:

$$\begin{aligned} \gamma (t):= {\left\{ \begin{array}{ll} {\mathbf{u}}(b+\delta - t, 0) &{} \mathrm{if}~ t\in (0,\delta ), \\ \gamma ^+(b+\delta - t)&{} \mathrm{if}~ t\in (\delta , b + \delta - a), \\ \gamma ^-(t + 2a - b - \delta )&{} \mathrm{if}~ t\in (b + \delta - a, 2(b-a) + \delta ), \\ {\mathbf{u}}(t+2a-b-\delta ,0) &{} \mathrm{if}~ t\in (2(b- a) + \delta , 2(b-a + \delta ) ). \end{array}\right. } \end{aligned}$$

We notice also that this construction can be done even if $\Omega$ and ${\mathbf{u}}$ satisfy condition I and $\gamma ^-(a)=\gamma ^+(a)$ (or, symmetrically, if $\gamma ^-(b)=\gamma ^+(b)$). In this case the sequence $({\mathbf{u}}_h)$ built in Proposition 7.1 would converge to ${\mathbf{u}}$ in $L^1(\mathrm{R};\mathbb {R}^2)$ and uniformly out of $J_{\mathbf{u}}$, and $\lim _{h \rightarrow + \infty } {\mathcal {A}}({\mathbf{u}}_h, \mathrm{R}) - \int _\mathrm{R}|{\mathcal {M}}(\nabla {\mathbf{u}})|\, \text{ d }t\, \text{ d }s \ge \overline{{\mathcal {A}}}^\infty _s({\mathbf{u}},\mathrm{R})=m(\mathrm{R};\varGamma ^-[{\mathbf{u}}],\varGamma ^+[{\mathbf{u}}])$.

Remark 7.3

(Relations with the vortex) Even if Proposition 7.1 has been inspired by the construction in [1, Lemma 5.3], it is worth to underline some important differences. When $\varOmega =B_R$ and ${\mathbf{u}}={{\mathbf{u}}_V}$, we could interpret the origin as a “collapsed” jump, and the radius $\{(t,0):\, t\in (-R,0)\}$ as the virtual jump. Differently from the case where $\varOmega$ and ${\mathbf{u}}$ satisfy condition II, the vortex map ${{\mathbf{u}}_V}$ (which belongs to $W^{1,p}(B_R; \mathbb S^1)$ for any $p\in [1,2)$) does not admit any limit as $(t,s)\rightarrow (0,0)$, $(t,s) \in B_R {\setminus } \{(t,0): t\in (-R,0)\}$. Intuitively if we would interpret the bound $\overline{{\mathcal {A}}}_s({{\mathbf{u}}_V}, B_R)\le 2\pi R$ in terms of area of semicartesian parametrizations, this lack of continuity of ${{\mathbf{u}}_V}$ at (0, 0) would force us to consider only surfaces having as trace on the plane $\{t=0\}\times \mathbb {R}^2_{(\xi , \eta )}$ the unit circumference, covered with the right orientation. We also notice that in [1, Lemma 5.3] the sequence $({\mathbf{u}}_h)$ is defined as the composition of ${{\mathbf{u}}_V}$ and a suitable retraction from $B_R$ to $B_R{\setminus } C_{\varepsilon _h}$, where $(C_{\varepsilon _h})$ is a decreasing sequence of circular sectors containing $\{(t,0): t\in (-R,0)\}$ and converging to it. In [1] the image of $C_{\varepsilon _h}$ through this retraction covers the whole of $B_R{\setminus } C_{\varepsilon _h}$, while in our case (for $C_{\varepsilon _h}$ now as in (7.4)) it is contained in $\partial C_{\varepsilon _h}$, see (7.5); this difference is due to the fact that, contrary to our case, the trace of the vortex map on the boundary of each circular sector is not continuous.

In the next example, we exhibit a map ${\mathbf{u}}$ for which the sequence built in Proposition 7.1 provides an upper bound that is lower than $m(D;\varGamma ^-, \varGamma ^+)$. This example, coupled with Theorem 1.4, shows that $\overline{{\mathcal {A}}}({\mathbf{u}},\Omega ) <\overline{{\mathcal {A}}}^\infty ({\mathbf{u}},\Omega )$.

Example 7.4

(Difference in the two relaxations) Let $\Omega$ be as in Proposition 7.1, and define¹¹

$$\begin{aligned} {\mathbf{u}}(t,s):= {\left\{ \begin{array}{ll} (1,0) &{} \text{ in } \{(t,s)\in \varOmega :\, s \ge 0\}, \\ (1,0) &{} \text{ in } \{(t,s)\in \varOmega :\, s < 0, t<a \text{ or } t>b\}, \\ \rho \Big (\cos \left( \theta (t)\right) ,\sin \left( \theta (t)\right) \Big ) + (1-\rho ,0) &{} \text{ in } \{(t,s)\in \varOmega :\, s < 0, a\le t \le b\}, \end{array}\right. } \end{aligned}$$

(7.7)

where

$$\begin{aligned} \rho > 2 (b+\delta -a), \end{aligned}$$

(7.8)

and $\theta :[a,b]\rightarrow [0,2\pi ]$ is defined in (2.11). We observe that $\gamma ^+$ is constant, $\gamma ^-$ covers once the circumference centered at $(1-\rho , 0)$ with radius $\rho$, and $\partial _t {\mathbf{u}}(t,0)=0$ for $t\in (b,b+\delta )$. In this case, formula (7.3) reads as

$$\begin{aligned} \overline{{\mathcal {A}}}_s({\mathbf{u}}, \varOmega )\le (b+\delta -a) \int _a^b |\dot{\gamma }^-|\,\hbox {d}t= (b+\delta - a)2\pi \rho . \end{aligned}$$

(7.9)

On the other hand, we have already observed in Example 2.13 that $a(\varGamma )\ge \pi \rho ^2$. Thus, from (7.8) we obtain that $a(\varGamma )$ is strictly greater than the right hand side of (7.9). Since in general $m(D;\varGamma ^-,\varGamma ^+)\ge a(\varGamma )$, see Remark 2.11, we have, using Theorem 1.4,

$$\begin{aligned} \overline{{\mathcal {A}}}_s({\mathbf{u}}, \varOmega )< m(D;\varGamma ^-[{\mathbf{u}}],\varGamma ^+[{\mathbf{u}}]) = \overline{{\mathcal {A}}}^\infty _s({\mathbf{u}}, \Omega ). \end{aligned}$$

(7.10)

Remark 7.5

(Joining two components of $J_{\mathbf{u}}$) Example 7.4 suggests that, if the jump set of a discontinuous map ${\mathbf{u}}: \varOmega \rightarrow \mathbb {R}^2$ is not connected, it could be convenient (as far as only the $L^1(\varOmega ; \mathbb {R}^2)$-convergence is involved) considering sequences $({\mathbf{u}}_h) \subset \mathrm{Lip}(\varOmega ; \mathbb {R}^2)$ converging to ${\mathbf{u}}$ in $L^1(\varOmega ;\mathbb {R}^2)$ and uniformly out of a connected curve containing $J_{\mathbf{u}}$. Let $\varOmega$ be as in Example 7.4 and let $\widetilde{\varOmega }$ be the union of $\varOmega$ and of its symmetrized with respect to the axis $\{t=c\}$, for some $c\in (b, b+\delta )$, see Fig. 6. Let us define ${\mathbf{u}}$ and $({\mathbf{u}}_h)$ in $\widetilde{\Omega }$ as in Example 7.4 for $(t,s)\in \varOmega$, and by reflection elsewhere in $\widetilde{\Omega }$. Then

$$\begin{aligned} \lim _{h\rightarrow +\infty }{\mathcal {A}}({\mathbf{u}}_h,\widetilde{\varOmega })- \int _{\widetilde{\varOmega }} |{\mathcal {M}}(\nabla {\mathbf{u}})|\,\text{ d }t\,\text{ d }s\le 4\pi \rho (c-a). \end{aligned}$$

(7.11)

If $\rho > 2 (c-a)$, the right hand side of (7.11) is smaller than $2 m(D;\varGamma ^-,\varGamma ^+)$ (where $\varGamma ^\pm$ are the graph of the traces of ${\mathbf{u}}$ on $(a,b)\times \{0\}$), that would be the bound obtained reasoning as in Proposition 5.1 in distinct neighborhoods of the two connected components of $J_{\mathbf{u}}$. Indeed, as observed in Example 7.4, $m(D;\varGamma ^-, \varGamma ^+)\ge a(\varGamma )\ge \pi \rho ^2> 4 \pi \rho (c-a)$. Moreover, if $J_{\mathbf{u}}$ is far from $\partial \widetilde{\varOmega }$, the right hand side of (7.11) is also smaller than the upper bound obtained by connecting each component of $J_{\mathbf{u}}$ with $\partial \widetilde{\varOmega }$, using the construction in Proposition 7.1.

Fig. 6

Remark 7.5. The set $\widetilde{\varOmega }$ is built by reflecting, with respect to the vertical axis in the figure, the set $\varOmega$ considered in Proposition 7.1. The map ${\mathbf{u}}: \widetilde{\varOmega }\rightarrow \mathbb {R}^2$ is defined again by reflecting the map in (7.7), so that $J_{\mathbf{u}}$ has two connected components. It is then possible to build a sequence $({\mathbf{u}}_h) \subset \mathrm{Lip}(\varOmega ; \mathbb {R}^2)$, again by reflection, converging to ${\mathbf{u}}$ in $L^1(\widetilde{\varOmega };\mathbb {R}^2)$ and uniformly out of the curve composed of $J_{\mathbf{u}}$ and of the bold dotted segment (the virtual jump) joining the two components of the jump. This sequence provides an upper bound for $\overline{{\mathcal {A}}}({\mathbf{u}}, \widetilde{\varOmega })$ lower than the one obtained by any sequence converging to ${\mathbf{u}}$ uniformly out of $J_{\mathbf{u}}$

7.2 Virtual jump starting from an interior point of $J_{\mathbf{u}}$

In this section we show, for a particular pair $(\varOmega ,{\mathbf{u}})$ satisfying condition II, how to build a sequence $({\mathbf{u}}_h)$ of maps converging to ${\mathbf{u}}$ in $L^1(\varOmega ;\mathbb {R}^2)$ and uniformly out of $J_\mathrm{ext}$, union of $J_{\mathbf{u}}$ and a virtual jump connecting an interior point of $J_{\mathbf{u}}$ and $\partial \varOmega$. Such phenomena indicate that the characterization of $\overline{{\mathcal {A}}}_s$ is involved, and justify, once more, the study of the functional $\overline{{\mathcal {A}}}^\infty _s$.

Let the bounded connected open set $\varOmega$ be such that $\{(\overline{t},s): s>0\}\cap {\varOmega }=\{\overline{t}\}\times (0,\delta )$ for some $\overline{t}\in (a,b)$. Without loss of generality we can suppose $\overline{t}=0$ and $a=-b$; we take $\varOmega := (-L,L)\times (-1,\delta )$, for $L > b>0$ and $\delta \in (0, 1)$, see Fig. 7. For $\rho > 2(\delta + b)$ we define ${\mathbf{u}}: \varOmega \rightarrow \mathbb {R}^2$ (similarly to Example 7.4) as

$$\begin{aligned} {\mathbf{u}}(t,s):= {\left\{ \begin{array}{ll} \displaystyle \rho \left( \cos \left( \frac{\pi }{b}t + \pi \right) , \sin \left( \frac{\pi }{b}t + \pi \right) \right) + (1-\rho ,0) &{} \mathrm{if}~ |t|\le b,\,\, s<0, \\ (1,0)\, &{} \mathrm{otherwise}. \end{array}\right. } \end{aligned}$$

We want to build a sequence $({\mathbf{u}}_h) \subset \mathrm{Lip}(\varOmega ;\mathbb {R}^2)$ converging to ${\mathbf{u}}$ in $L^1(\varOmega ;\mathbb {R}^2)$ and uniformly out of

$$\begin{aligned} J_\mathrm{ext}:= J_{\mathbf{u}}\cup \big (\{0\} \times (0,\delta )\big ), \end{aligned}$$

where $J_\mathrm{ext}{\setminus } J_{\mathbf{u}}= \{0\} \times (0,\delta )$ takes the role of the virtual jump (see (7.13) below) and such that

$$\begin{aligned} \lim _{h \rightarrow + \infty }{\mathcal {A}}({\mathbf{u}}_h,\varOmega ) = \int _\varOmega |{\mathcal {M}}(\nabla {\mathbf{u}})|\,\text{ d }t\,\text{ d }s + 2\pi \rho (\delta + b). \end{aligned}$$

(7.12)

To this purpose, for $\varepsilon \in (0, \min \{\delta /2, b/2\})$ set (see Fig. 7)

$$\begin{aligned} \mathbb {T}_\varepsilon:= & {} \{(t,s)\in \varOmega : \,|t|<b, s>0,\, \mathrm{dist}((t,s), J_\mathrm{ext})< \varepsilon \}, \\ \mathbb {T}^-_\varepsilon:= & {} \mathbb {T}_\varepsilon \cap \{t < 0\}, \qquad \qquad \mathbb {T}^+_\varepsilon := \mathbb {T}_\varepsilon \cap \{t> 0\}. \end{aligned}$$

The definitions will be given on $\mathbb {T}^+_\varepsilon$ and next extended on the whole of $\mathbb {T}_\varepsilon$ by reflection with respect to the s-axis. Let us parametrize the (closure of the) curve $\Omega \cap \{t>0\} \cap \partial \mathbb {T}_\varepsilon$ by the arc-length parametrization $\lambda _\varepsilon \in \mathrm{Lip}([0, \delta + 2 b -\varepsilon ];\mathbb {R}^2 )$ with $\lambda _\varepsilon (0)= (\varepsilon , \delta )$ and $\lambda _\varepsilon (\delta + 2 b -\varepsilon )=(0,0)$. Let us also parametrize the (closure of the) set $J_\mathrm{ext}^+:= (\{0\}\times [0,\delta )) \cup ([0,b)\times \{0\})$ by the arc-length parametrization $\tilde{\alpha } \in \mathrm{Lip} ([0,\delta + b]; \mathbb {R}^2)$, with $\tilde{\alpha }(0)=(0,\delta )$ and $\tilde{\alpha }(b+\delta )=(b,0)$.

Fig. 7

In the left picture the set $\varOmega$ and, in bold, the jump set $(-b,b)\times \{0\}$ of ${\mathbf{u}}$. The virtual jump $\{0\}\times (0,\delta )$ is represented by a dotted segment, while the set $\mathbb {T}_\varepsilon$ is in gray. In the right picture we show the details of $\mathbb {T}_\varepsilon ^+$ and we provide a representation of the (p, d) coordinates in $\mathbb {T}_\varepsilon ^+$ used to define the retraction $r_\varepsilon$. Every point of $\mathbb {T}_\varepsilon ^+$ belongs to exactly one of the segments (horizontal, oblique, or vertical) represented in the picture; its d-coordinate is its distance from the point of the same segment belonging either to the jump or to the virtual jump; points belonging to the same segment have the same p-coordinate

On $\mathbb {T}_\varepsilon ^+$ we consider the change of coordinates $(p,d): \mathbb {T}_\varepsilon ^+ \rightarrow S_\varepsilon ^+$ defined as follows¹²:

• $p(t,s)\in (0,\delta +b)$ is the image through $\tilde{\alpha }^{-1}$ of the endpoint on $J_\mathrm{ext}^+$ of the segment represented in Fig. 7 passing through (t, s);

• d(t, s) is the distance between (t, s) and $\tilde{\alpha }(p(t,s))$.

The set $S^+_\varepsilon$ is therefore $\{(p,d)\in \mathbb {R}^2: p\in (0,\delta + b), d\in (0,d_{\max }(p))\}$, where

$$\begin{aligned} d_{\max }(p)= {\left\{ \begin{array}{ll} \varepsilon \, &{} \mathrm{if}~ p\in (0,\delta -\varepsilon ], \\ \sqrt{\varepsilon ^2 + (p - \delta + \varepsilon )^2}\, &{} \mathrm{if}~ p \in (\delta - \varepsilon , \delta ], \\ \sqrt{\varepsilon ^2 + (-p + \delta + \varepsilon )^2}\, &{} \mathrm{if}~ p \in (\delta , \delta + \varepsilon ], \\ \varepsilon \, &{} \mathrm{if}~ p\in (\delta +\varepsilon , \delta + b). \end{array}\right. } \end{aligned}$$

Now, we define a function $\ell _\varepsilon :S^+_\varepsilon \rightarrow (0,\delta + 2b -\varepsilon )$, linear on each segment $\{p\}\times (0,d_{\max }(p))$ and such that:

• for $p\in (0,\delta -\varepsilon ]$, $\ell _\varepsilon (p,0)=\delta + 2b - \varepsilon$ and $\ell _\varepsilon (p,d_{\max }(p))=p$;

• for $p\in (\delta -\varepsilon ,\delta ]$, $\ell _\varepsilon (p,0)=\delta + 2b - \varepsilon$ and $\ell _\varepsilon (p,d_{\max }(p))=\delta - \varepsilon$;

• for $p\in (\delta ,\delta + \varepsilon ]$, $\ell _\varepsilon (p,0)=2 \delta + 2b - \varepsilon - p$ and $\ell _\varepsilon (p,d_{\max }(p))=\delta - \varepsilon$;

• for $p\in (\delta + \varepsilon , \delta + b)$, $\ell _\varepsilon (p,0)=2 \delta + 2b - \varepsilon - p$ and $\ell _\varepsilon (p,d_{\max }(p))=p - 2\varepsilon$.

Thus we can define the retraction $r_\varepsilon = (r_{\varepsilon ,1}, r_{\varepsilon ,1}): \varOmega \rightarrow \varOmega {\setminus } \mathbb {T}_\varepsilon$ as:

• $r_{\varepsilon }:=\mathrm{id}$ on $\varOmega {\setminus } \mathbb {T}_\varepsilon$;

• $r_\varepsilon (t,s):=\lambda _\varepsilon \left( \ell _\varepsilon \big (p(t,s),d(t,s)\big )\right)$ if $(t,s)\in \mathbb {T}_\varepsilon ^+$;

• $r_\varepsilon (0,s):=(0,0)$ if $s\in [0,\delta )$;

• $r_\varepsilon (t,s):=(-r_{\varepsilon ,1}(-t,s), r_{\varepsilon ,2}(-t,s))$ if $(t,s)\in \mathbb {T}_\varepsilon ^-$.

In words, on $\mathbb {T}^+_\varepsilon$ the map $r_\varepsilon$ sends each segment in Fig. 7 into $\partial \mathbb {T}_\varepsilon \cap \{t>0\}$ in such a way that:

• if both the endpoints of the segment lie on $\partial \mathbb {T}_\varepsilon \cap \{t>0\}$ (namely, the vertical segments and the oblique segments below the diagonal), the image of the segment is the portion of $\partial \mathbb {T}_\varepsilon \cap \{t>0\}$ bounded by the two endpoints;

• if only one of the endpoints of the segment lies on $\partial \mathbb {T}_\varepsilon \cap \{t>0\}$, the image of the segment is the portion of $\partial \mathbb {T}_\varepsilon \cap \{t>0\}$ bounded by that endpoint and (0, 0).

Let $A_{\varepsilon } := r_{\varepsilon }^{-1}(J_{\mathbf{u}}) \subset T_\varepsilon$;. the image of $A_\varepsilon \cap \mathbb {T}^+_\varepsilon$ through the coordinate change (p, d) is the subset of $S^+_\varepsilon$ given by $\{(p,d):\, p\in (0,\delta +b),\,d\in (0,d_{J_{\mathbf{u}}}(p)]\}$, where

$$\begin{aligned} d_{J_{\mathbf{u}}}(p):= {\left\{ \begin{array}{ll} \frac{b}{\delta + 2 b - \varepsilon - p}\varepsilon \, &{} \mathrm{if}~ p \in (0,\delta - \varepsilon ], \\ \frac{d_{\max }(p)}{2}\, &{} \mathrm{if}~ p \in (\delta - \varepsilon , 0], \\ \frac{\delta + b - p}{\delta + 2b - p}d_{\max }(p) \, &{} \mathrm{if}~ p \in (0,\delta + \varepsilon ], \\ \frac{\delta + b - p}{2\delta + 2 b - 2 p + \varepsilon }\varepsilon \, &{} \mathrm{if}~ p \in (\delta + \varepsilon , \delta + b). \end{array}\right. } \end{aligned}$$

We are now in the position to define $({\mathbf{u}}_h) \subset \mathrm{Lip}(\varOmega ; \mathbb {R}^2)$ as

$$\begin{aligned} {\mathbf{u}}_h(t,s):= {\left\{ \begin{array}{ll} {\mathbf{u}}(r_{\varepsilon _h}(t,s)) &{} \mathrm{if}~ (t,s) \in \varOmega {\setminus } A_{\varepsilon _h}, \\ \gamma ^-[{\mathbf{u}}](r_{\varepsilon _h, 1}(t,s)) &{} \mathrm{if}~ (t,s) \in A_{\varepsilon _h}, \end{array}\right. } \end{aligned}$$

(7.13)

where $(\varepsilon _h)\subset (0,\min \{\delta /2, b/2\})$ is an infinitesimal sequence.

Since $\partial _2 {\mathbf{u}}=0$ almost everywhere,¹³ we have

$$\begin{aligned} {\mathcal {A}}({\mathbf{u}}_h,\varOmega )=\int _\varOmega \sqrt{1 + |\partial _1 {\mathbf{u}}(r_{\varepsilon _h})|^2 \left( (\partial _t r_{\varepsilon _h})^2 + (\partial _s r_{\varepsilon _h})^2\right) } ~\hbox {d}t\,\hbox {d}s. \end{aligned}$$

We observe also that $r_{\varepsilon _h}( \mathbb {T}_{\varepsilon _h}{\setminus } A_{\varepsilon _h}) \subseteq \Omega \cap (\partial \mathbb {T}_{\varepsilon _h} {\setminus } J_{\mathbf{u}})$, where also $\partial _1 {\mathbf{u}}=0$.

Let us compute the area of the graph of the map ${\mathbf{u}}_h$ on $\mathbb {T}_{\varepsilon _h}^+ \cap \{s>{\varepsilon _h}\}$, on $\mathbb {T}_{\varepsilon _h}^+ \cap \{t>{\varepsilon _h}\}$ and on $(0,{\varepsilon _h})\times (0,{\varepsilon _h})$, separately.

On $\mathbb {T}_{\varepsilon _h}^+ \cap \{s>{\varepsilon _h}\}$ we have that $p(t,s)= \delta - s$ and $d(t,s)=t$. Thus

$$\begin{aligned} A_{\varepsilon _h}\cap ( \mathbb {T}_{\varepsilon _h}^+ \cap \{s>{\varepsilon _h}\}) =\left\{ (t,s)\in \mathbb {T}^+_{\varepsilon _h}:\,\, s\in (\varepsilon ,\delta ), \, t\in \left( 0, \frac{b\varepsilon _h}{2b - \varepsilon +s}\right) \right\} , \end{aligned}$$

and for (t, s) in this set, ${\mathbf{u}}_h(t,s)= \gamma ^-[{\mathbf{u}}]\left( \frac{2b-\varepsilon _h+s}{\varepsilon _h} t\right)$. Hence

$$\begin{aligned} \begin{aligned}&{\mathcal {A}}({\mathbf{u}}_h, \mathbb {T}^+_{\varepsilon _h} \cap \{s>{\varepsilon _h}\}) = {\mathcal {A}}({\mathbf{u}}_h, ( \mathbb {T}^+_{\varepsilon _h} {\setminus } A_{\varepsilon _h}) \cap \{s>{\varepsilon _h}\}) + {\mathcal {A}}({\mathbf{u}}_h, A_{\varepsilon _h}\cap \mathbb {T}^+_{\varepsilon _h} \cap \{s>{\varepsilon _h}\})\\&\quad = \int _{{\varepsilon _h}}^\delta \int _{\frac{b{\varepsilon _h}}{2b + s - {\varepsilon _h}}}^{\varepsilon _h} 1 \,\hbox {d}t\,\hbox {d}s + \int _{{\varepsilon _h}}^\delta \int _{0}^{\frac{b{\varepsilon _h}}{2b + s - {\varepsilon _h}}} \sqrt{1 + \left| \dot{\gamma }^- \left( \frac{2b + s -{\varepsilon _h}}{{\varepsilon _h}}t\right) \right| ^2 \left( \frac{(2b + s-{\varepsilon _h})^2}{{\varepsilon _h}^2} + \frac{t^2}{{\varepsilon _h}^2}\right) }\,\hbox {d}t \, \hbox {d}s\\&\quad = {\varepsilon _h}\int _{\varepsilon _h}^\delta \left( \frac{b+s-{\varepsilon _h}}{2b + s - {\varepsilon _h}}\right) \,\hbox {d}s + \int _{{\varepsilon _h}}^\delta \int _{0}^b \sqrt{\mathcal {O}({\varepsilon _h}^2) + |\dot{\gamma }^-(\tau )|^2 (1 + \mathcal {O}({\varepsilon _h}^2))} ~ \hbox {d}\tau \,\hbox {d}s \\&\overset{h \rightarrow +\infty }{\longrightarrow } \delta \int _0^a |{\dot{\gamma }}^-(\tau )| ~ \hbox {d}\tau = \pi \rho \delta . \end{aligned} \end{aligned}$$

On $\mathbb {T}^+_{\varepsilon _h}\cap \{t>\varepsilon _h\}$ we have $p(t,s)= \delta + t$ and $d(t,s)= s$. Thus

$$\begin{aligned} A_{\varepsilon _h}\cap ( \mathbb {T}^+_{\varepsilon _h}\cap \{t>\varepsilon _h\}) = \left\{ (t,s)\in \mathbb {T}^+_\varepsilon : \,\,t \in (\varepsilon _h, b),\,s\in \left( 0, \frac{(b-t)\varepsilon _h}{2b - 2t + \varepsilon _h}\right) \right\} , \end{aligned}$$

and for (t, s) in this set, ${\mathbf{u}}_h(t,s)= \gamma ^-[{\mathbf{u}}]\left( \frac{2b-2t + \varepsilon _h}{\varepsilon _h}s + t \right)$. Hence

$$\begin{aligned} \begin{aligned}&{\mathcal {A}}({\mathbf{u}}_h, \mathbb {T}^+_{\varepsilon _h} \cap \{t>\varepsilon _h\})= {\mathcal {A}}({\mathbf{u}}_h, ( \mathbb {T}^+_{\varepsilon _h} {\setminus } A_{\varepsilon _h}) \cap \{t>{\varepsilon _h}\}) + {\mathcal {A}}({\mathbf{u}}_h, A_{\varepsilon _h}\cap \mathbb {T}^+_{\varepsilon _h} \cap \{t>{\varepsilon _h}\}) \\&\quad = \int _{{\varepsilon _h}}^b \int _{\frac{(b-t){\varepsilon _h}}{2b -2t + {\varepsilon _h}}}^{\varepsilon _h} 1 \,\hbox {d}t\,\hbox {d}s+ \int _{\varepsilon _h}^b\int _{0}^{\frac{(b-t){\varepsilon _h}}{2b -2t + {\varepsilon _h}}} \sqrt{1 + \left| \dot{\gamma }^- \left( t+ \frac{2b-2t+{\varepsilon _h}}{{\varepsilon _h}}s\right) \right| ^2 \left( \left( 1 - \frac{2s}{{\varepsilon _h}}\right) ^2 +\frac{(2b-2t+{\varepsilon _h})^2}{{\varepsilon _h}^2}\right) }\,\hbox {d}s \, \hbox {d}t \\&\quad ={\varepsilon _h}\int _{\varepsilon _h}^b\left( \frac{b-t+{\varepsilon _h}}{2b - 2t + {\varepsilon _h}}\right) \,\hbox {d}t + \int _{\varepsilon _h}^b \int _0^b \sqrt{\mathcal {O}({\varepsilon _h}^2) + |\dot{\gamma }^-(\tau )|^2(1 + \mathcal {O}({\varepsilon _h}^2))}\,\hbox {d}\tau \,\hbox {d}t \\&\overset{h \rightarrow +\infty }{\longrightarrow } b \int _0^b |{\dot{\gamma }}^-(\tau )|\,\hbox {d}\tau = \pi \rho b. \end{aligned} \end{aligned}$$

In order to estimate the area of the graph of ${\mathbf{u}}_h$ on $(0,\varepsilon _h)\times (0,\varepsilon _h)$, is it enough to notice that $|\nabla r_{\varepsilon _h,1}|=\mathcal {O}(\varepsilon _h^{-1})$. Therefore

$$\begin{aligned} \begin{aligned} {\mathcal {A}}({\mathbf{u}}_h, (0,\varepsilon _h)\times (0,\varepsilon _h))&= \int _0^{\varepsilon _h}\int _0^{\varepsilon _h}\sqrt{1 + |\partial _1 {\mathbf{u}}(r_{\varepsilon _h})|^2 \left( (\partial _t r_{\varepsilon _h})^2 + (\partial _s r_{\varepsilon _h})^2\right) }\,\hbox {d}t\,\hbox {d}s \\ {}&\le \varepsilon ^{-1}_h \int _0^{\varepsilon _h}\int _0^{\varepsilon _h}\sqrt{\mathcal {O}(\varepsilon _h^2) + \mathcal {O}(1)}\overset{h\rightarrow +\infty }{\longrightarrow }0. \end{aligned} \end{aligned}$$

Since by symmetry $\displaystyle \lim _{h \rightarrow + \infty }{\mathcal {A}}({\mathbf{u}}_h, \mathbb {T}_{\varepsilon _h}^+) = \displaystyle \lim _{h \rightarrow +\infty }{\mathcal {A}}({\mathbf{u}}_h, \mathbb {T}_{\varepsilon _h}^-)$, (7.12) follows. This implies that $\overline{{\mathcal {A}}}_s({\mathbf{u}}, \varOmega ) \le 2 \pi \rho (\delta + b)$. Due to our choice of $\rho$, we conclude, as in Example 7.4, that $\overline{{\mathcal {A}}}_s({\mathbf{u}}, \varOmega ) <m(D;\varGamma ^-[{\mathbf{u}}], \varGamma ^+[{\mathbf{u}}])$.

8 Non-subadditivity of $\overline{{\mathcal {A}}}$

Let us suppose that $\varOmega$ and ${\mathbf{u}}$ satisfy condition I, and choose a map ${\mathbf{u}}$ of the form

$$\begin{aligned} {\mathbf{u}}(t,s):= {\left\{ \begin{array}{ll} (f(t),0) &{} \mathrm{if}~ (t,s) \in \mathrm{R}^-, \\ (f(t),1) &{} \mathrm{if}~ (t,s) \in \mathrm{R}^+, \end{array}\right. } \end{aligned}$$

(8.1)

where $f\in \mathrm{Lip}([a,b])$ is a piecewise $\mathcal C^1$-function¹⁴; for simplicity, we fix $f(a)=0$. Clearly $\gamma ^-(t)=(f(t), 0)$ and $\gamma ^+(t)=(f(t),1)$.

The aim of this section is to prove the following result.

Theorem 8.1

(Non-subadditivity of $\overline{{\mathcal {A}}}({\mathbf{u}},\cdot )$) Let ${\mathbf{u}}$ be as in (8.1) for a non-constant function f. Then $\overline{{\mathcal {A}}}({\mathbf{u}}, \cdot )$ is not subadditive.

Remark 8.2

It is worth to recall that in [1] it is proven that if f is constant then $\overline{{\mathcal {A}}}_s({\mathbf{u}},\cdot )=|D^s{\mathbf{u}}|(\cdot )$, and thus $\overline{{\mathcal {A}}}({\mathbf{u}}, \cdot )$ is subadditive.

In order to prove Theorem 8.1, we need some intermediate results: Proposition 8.3, that provides an estimate from above of $\overline{{\mathcal {A}}}_s({\mathbf{u}}, \mathrm{R})$, in terms of the area of a suitable semicartesian parametrization (see Fig. 8); Proposition 8.4, where we show that, if $\overline{{\mathcal {A}}}({\mathbf{u}},\cdot )$ were subadditive, than $\overline{{\mathcal {A}}}_s({\mathbf{u}}, \cdot )$ would be forced to coincide with $|D^s{\mathbf{u}}|(\cdot )$ (see (8.3)); Proposition 8.6, that characterizes suitable “vertical” bidimensional currents in $\mathbb {R}^4$, whose mass is controlled from above.

We stress that from Proposition 8.4 it follows that there exist coplanar curves $\varGamma ^\pm$ such that the image of an area-minimizing semicartesian parametrization (if it exists) spanning $\varGamma ^- \cup \varGamma ^+$ is not planar, see Remark 8.5: this is a consequence of the fact that the area of the rectangle E bounding $\varGamma ^-$ and $\varGamma ^+$ in Fig. 8 can be larger than the sum of the area of its orthogonal projection on the $t\eta$-plane and the areas of the two triangles ${\mathcal {T}}_0$ and ${\mathcal {T}}_1$ (see inequality (8.4)).

Proposition 8.3

Let ${\mathbf{u}}$ be as in (8.1). Then

$$\begin{aligned} \overline{{\mathcal {A}}}({\mathbf{u}},\mathrm{R})\le \int _\mathrm{R}|{\mathcal {M}}(\nabla {\mathbf{u}})|\,dt\,ds + |D^s {\mathbf{u}}|(\mathrm{R}) + \mathrm{lip}(f) (b-a)^2. \end{aligned}$$

Proof

Since $|{\mathbf{u}}^+(t,0)- {\mathbf{u}}^-(t,0)|=1$ for any $t\in (a,b)$, we have

$$\begin{aligned} |D^s {\mathbf{u}}|(\mathrm{R}) = b-a = {\mathcal {H}}^2(V), \end{aligned}$$

where V is the rectangle $V:=[a,b]\times \{0\}\times [0,1] \subset \mathbb {R}^3_{(t,\xi ,\eta )}$ (see Fig. 8). Let

$$\begin{aligned} {\mathcal {T}}_0:=\left\{ (t,\xi ,0):\,\, t\in (a,b), \ \xi \in (0,f(t)) \text{ if } 0\le f(t),\,\,\xi \in (f(t),0) \text{ otherwise }\right\} \end{aligned}$$

and ${\mathcal {T}}_1:={\mathcal {T}}_0 + (0,0,1)$. We observe that, since $f(a)=0$,

$$\begin{aligned} \mathcal {H}^2 ({\mathcal {T}}_0)=\mathcal {H}^2 ({\mathcal {T}}_1)\le \frac{\mathrm{lip}(f)}{2}(b-a)^2, \end{aligned}$$

and hence, if

$$\begin{aligned} \Sigma :={\mathcal {T}}_0\cup V\cup {\mathcal {T}}_1, \end{aligned}$$

(8.2)

we have

$$\begin{aligned} {\mathcal {H}}^2(\Sigma ) \le |D^s {\mathbf{u}}|(\mathrm{R}) + \mathrm{lip}(f) (b-a)^2. \end{aligned}$$

Fig. 8

The curves $\varGamma ^-=\{(t,f(t),0) : t \in [a,b]\}$, $\varGamma ^+=\{(t,f(t),1) : t \in [a,b]\}$ for a linear non-constant function f, the rectangle V, and the sets ${\mathcal {T}}_0$, and ${\mathcal {T}}_1$ (in this case two triangles at different heights). The broken dotted curve is the image through the map $\varPhi$ of the vertical segment $\mathrm{R}\cap \{t = \overline{t}\}$

Recalling Proposition 3.1, the result follows if we prove that $\Sigma$ can be parametrized by an injective map $\varPhi \in \mathrm{semicart}(\mathrm{R};\varGamma ^-,\varGamma ^+)$. This is true, by considering for example the map $\varPhi (t,s):=(t, \phi (t,s))$, with $\phi \in H^1(\mathrm{R};\mathbb {R}^2)$ defined by

$$\begin{aligned} \phi (t,s):= {\left\{ \begin{array}{ll} \frac{1-s}{2/3} (0,1) + \frac{s - 1/3}{2/3} \gamma ^+(t) &{} \text{ in } \mathrm{R}\cap \{ s\ge 1/3\}, \\ \frac{s + 1/3}{2/3}(0,1) &{} \text{ in } \mathrm{R}\cap \{ -1/3 \le s < 1/3\}, \\ -\frac{s + 1/3}{2/3}\gamma ^-(t) &{} \text{ in } \mathrm{R}\cap \{ -1 < s < - 1/3\}, \end{array}\right. } \end{aligned}$$

which satisfies $\varPhi (R)=\Sigma$. $\square$

Proposition 8.4

(Subadditivity and coincidence with $\vert D^s {\mathbf{u}}\vert$) Let ${\mathbf{u}}$ be as in (8.1) for a non-constant function f. If the functional $\overline{{\mathcal {A}}}({\mathbf{u}}, \cdot )$ were subadditive, then

$$\begin{aligned} \overline{{\mathcal {A}}}_s({\mathbf{u}}, \mathrm{R})= |D^s {\mathbf{u}}|(\mathrm{R}). \end{aligned}$$

(8.3)

Proof

Fix $\delta \in (0,(b-a)/2)$ and let $N(\delta )\in \mathbb {N}$ be such that $a+N(\delta )\delta < b \le a + (N(\delta )+1)\delta$. Define

$$\begin{aligned} \begin{array}{ll} \mathrm{R}_i :=[(a + i\delta , a+ (i+1)\delta )\cap (a,b)]\times (-1,1), &{} \text{ for }\;i = 0, \dots , N(\delta ), \\ \mathrm{P}_i:=[(a + i\delta - \delta ^2, a + i\delta + \delta ^2)\cap (a,b)]\times (-1,1), &{} \text{ for }\;i = 1, \dots , N(\delta ). \end{array} \end{aligned}$$

From Proposition 8.3, applied with $\mathrm{R}_i$ and $\mathrm{P}_i$ in place of $\mathrm{R}$, it follows

$$\begin{aligned} \begin{aligned} \overline{{\mathcal {A}}}({\mathbf{u}},\mathrm{R}_i)&\le \int _{\mathrm{R}_i} |{\mathcal {M}}(\nabla {\mathbf{u}})|\,\text{ d }t\,\text{ d }s + |D^s {\mathbf{u}}|(\mathrm{R}_i)+ \mathrm{lip}(f) \delta ^2, \\ \overline{{\mathcal {A}}}({\mathbf{u}},\mathrm{P}_i)&\le \int _{\mathrm{P}_i} |{\mathcal {M}}(\nabla {\mathbf{u}})|\,\text{ d }t\,\text{ d }s + |D^s {\mathbf{u}}|(\mathrm{P}_i)+ 4\mathrm{lip}(f) \delta ^4. \end{aligned} \end{aligned}$$

If $\overline{{\mathcal {A}}}({\mathbf{u}},\cdot )$ were subadditive, we would get

Since by [1, Theorem 3.7] we have

the thesis follows letting $\delta \rightarrow 0^+$. $\square$

Remark 8.5

(Non-planarity) Let us consider a map as in (8.1), for a linear function $f(t)= c(t-a)$, $c>0$ (Fig. 8). In this case the curves $\varGamma ^\pm$ are coplanar. The map $\varPsi \in \mathrm{semicart}(\mathrm{R}; \varGamma ^-, \varGamma ^+)$ defined by

$$\begin{aligned} \varPsi (t,s):=\left( t, c(t-a), \frac{s+1}{2}\right) , \qquad t\in (a,b), \,\,\, s\in (-1,1), \end{aligned}$$

parametrizes the rectangle E bounded by $\varGamma ^-$ and $\varGamma ^+$ and the two vertical segments $\{(a,0,\eta ):\,\eta \in (0,1)\}$, and $\{(b,c(b-a),\eta ):\,\eta \in (0,1)\}$. We have

$$\begin{aligned} \mathcal H^2(E) = (b-a)\sqrt{1 + c^2}=\int _\mathrm{R}|\partial _t \varPsi \wedge \partial _s \varPsi | \,\text{ d }t\,\text{ d }s. \end{aligned}$$

On the other hand, the sets ${\mathcal {T}}_0$ and ${\mathcal {T}}_1$ built in Proposition 8.4 are two triangles of area $\frac{c(b-a)^2}{2}$, so that ${\mathcal {H}}^2(\Sigma )= c (b-a)^2 + (b-a)$, where $\Sigma$ is defined in (8.2). A simple computation shows that

$$\begin{aligned} {\mathcal {H}}^2(E)>{\mathcal {H}}^2(\Sigma ) \end{aligned}$$

(8.4)

provided that $b-a \in \left( 0, \frac{\sqrt{1+c^2}-1}{c}\right)$. Interestingly, this implies that an area-minimizing semicartesian surface (if it exists, recall the beginning of Remark 2.11) spanning two coplanar curves is not necessarily planar.

The next proposition is a modification of [1, Lemma 4.8]. We refer to [1] and [9] for all notations and results concerning cartesian currents.

Proposition 8.6

(Mass lower bound of cartesian two-currents) Let $T=\tau (S_T,\theta _T,\zeta _T)$ be a 2-dimensional integer rectifiable current with bounded support in $U:=\mathrm{R}\times \mathbb {R}^2\subset \mathbb {R}^2_{(t,s)}\times \mathbb {R}^2_{(\xi ,\eta )}$. Denote by $p: U \rightarrow \mathbb {R}^2_{(t,s)}$ the orthogonal projection. Suppose that

• (i) $\mathcal {L}^2(p(S_T))=0$,

• (ii) $\partial T= [[\{(t,0,f(t),1)\}_{t\in (a,b)}]]-[[\{(t,0,f(t),0)\}_{t\in (a,b)}]]$ in U, where $f\in \mathrm{Lip}([a,b])$ is piecewise $\mathcal C^1$.

Then

$$\begin{aligned} f \hbox {non-constant} \quad \Rightarrow \quad {\mathbf {M}}_U(T)> \int _a^b |(f(t),1) - (f(t), 0)|\,\hbox {d}t= b-a. \end{aligned}$$

Proof

Assume by contradiction that ${\mathbf {M}}_U(T)\le b-a$. Let $\pi :U\rightarrow \mathrm{R}\times \mathbb {R}_\eta$ and $q:\mathrm{R}\times \mathbb {R}_\eta \rightarrow \mathrm{R}$ be the orthogonal projections, so that $p = q \circ \pi$. Since T has bounded support in U, it follows that $\partial (\pi _\sharp T)=\pi _\sharp (\partial T)$ (see [9, Sec2.3]) and thus, from assumption (ii), $\partial (\pi _\sharp T)=[[\{(t,0,1)\}_{t\in (a,b)}]]-[[\{(t,0,0)\}_{t\in (a,b)}]]=: [[(a,0),(b,0)]]\times [[1]]- [[(a,0),(b,0)]]\times [[0]]$ in $\mathrm{R}\times \mathbb {R}_\eta$. Since T is rectifiable, also $\pi _\sharp T$ is rectifiable, and $\pi _\sharp T=\tau (S_{\pi _\sharp T}, \theta _{\pi _\sharp T}, \zeta _{\pi _\sharp T})$. Moreover, $S_{\pi _\sharp T}\subseteq \pi (S_T)$ and thus, applying the projection q and recalling assumption (i), we get $\mathcal {L}^2(q(S_{\pi _\sharp T}))=0$. Applying the one-codimensional result in [1, Lemma 4.7], we deduce that $\pi _\sharp T = [[(a,0), (b,0)]]\times [[0,1]]$. In particular $S_{\pi _\sharp T}=(a,b)\times \{0\}\times \{0\}\times [0,1]$, $\theta _{\pi _\sharp T} = 1$, and $\zeta _{\pi _\sharp T}= (1,0,0,0)\wedge (0,0,0,1)$.

Now, we use the assumption on the mass of the current T, obtaining

$$\begin{aligned} b-a = {\mathcal {H}}^2(S_{\pi _\sharp T})\le {\mathcal {H}}^2(\pi (S_T))\le {\mathcal {H}}^2(S_T)\le {\mathbf {M}}_U(T)\le b-a; \end{aligned}$$

hence the above inequalities are indeed equalities and in particular

$$\begin{aligned} \pi (S_T)\simeq (a,b)\times \{0\}\times \{0\}\times [0,1] \end{aligned}$$

in the sense of ${\mathcal {H}}^2$. Moreover, since ${\mathcal {H}}^2(\pi (S_T))= {\mathcal {H}}^2(S_T) = {\mathbf {M}}_U(T)$, it follows that $\theta _T = 1$ $\mathcal H^2$-almost everywhere on $S_T$, and $\zeta _T = e_t\wedge \varepsilon _\eta$, where $e_t= (1,0,0,0)$ and $\varepsilon _\eta = (0,0,0,1)$, see [1, Lemma 4.8].

Thus, if we write any smooth 2-form $\omega$ compactly supported in U as

$$\begin{aligned} \omega := \omega ^{t,s}dt\wedge ds + \omega ^{t,\xi } dt\wedge d\xi + \omega ^{t,\eta } dt\wedge d\eta + \omega ^{s,\xi } ds\wedge d\xi + \omega ^{s, \eta }ds\wedge d\eta + \omega ^{\xi ,\eta } d\xi \wedge d\eta , \end{aligned}$$

we have

$$\begin{aligned} T(\omega )=\int _{S_T}\omega ^{t,\eta }. \end{aligned}$$

Let $\alpha := \alpha ^t \hbox {d}t + \alpha ^s \hbox {d}s + \alpha ^\xi \hbox {d}\xi + \alpha ^\eta \hbox {d} \eta$ be a smooth 1-form compactly supported in U. Then

$$\begin{aligned} \partial T(\alpha )=T(d\alpha )= \int _{S_T} (\partial _t \alpha ^\eta - \partial _\eta \alpha ^t). \end{aligned}$$

(8.5)

On the other hand, from assumption (ii) it follows

$$\begin{aligned} \partial T(\alpha )=\int _a^b \left[ \alpha ^t(t,0,f(t), 1) - \alpha ^t(t,0,f(t), 0) \right] \nonumber \\ \quad + f'(t)\left[ \alpha ^\xi (t,0,f(t), 1) - \alpha ^\xi (t,0,f(t), 0) \right] \,\text{ d }t. \end{aligned}$$

(8.6)

Now, we choose the $1-$form $\alpha$ such that $\alpha ^t = 0$, $\alpha ^\eta =0$, so that the right hand side of (8.5) vanishes, and (8.6) reduces to

$$\begin{aligned} \partial T(\alpha )= \int _a^b f'(t)\left[ \alpha ^\xi (t,0,f(t), 1) - \alpha ^\xi (t,0,f(t), 0) \right] \,\text{ d }t. \end{aligned}$$

Since f is not constant and piecewise $\mathcal C^1$, there exists a non-empty open interval $I\subset (a,b)$ where $f'$ is either positive or negative. It is then sufficient to choose $\alpha ^\xi$ such that $\alpha ^\xi (t,0,f(t), 1) \ne \alpha ^\xi (t,0,f(t), 0)$ for any $t \in I$, to obtain that the right hand side of (8.6) is nonzero, which is a contradiction. $\square$

Now, we are in a position to prove Theorem 8.1

Proof

Let us suppose by contradiction that $\overline{{\mathcal {A}}}({\mathbf{u}};\cdot )$ is subadditive. As a consequence of Proposition 8.4 and [1, Lemma 3.3], we can select a sequence $({\mathbf{u}}_h)\subset \mathcal {C}^1(\mathrm{R}; \mathbb {R}^2)$ converging to ${\mathbf{u}}$ in $L^1(\mathrm{R}; \mathbb {R}^2)$ and bounded in $L^\infty (\mathrm{R}; \mathbb {R}^2)$, and such that

$$\begin{aligned} {\mathcal {A}}({\mathbf{u}}_h,\mathrm{R})\rightarrow \overline{{\mathcal {A}}}({\mathbf{u}}, \mathrm{R})=\int _{\mathrm{R}} |{\mathcal {M}}(\nabla {\mathbf{u}})|\,\text{ d }t\,\text{ d }s + |D^s {\mathbf{u}}|(\mathrm{R}) \qquad \text{ as } h\rightarrow +\infty . \end{aligned}$$

From [1, Theorem 2.6 and Remark 2.4] it follows that the sequence of the graphs $[[G_{{\mathbf{u}}_h}]]$ (as currents) of the maps ${\mathbf{u}}_h$ converges weakly in the distributional sense to a cartesian (integer rectifiable) 2-current $T=\tau (S, \theta , \zeta )$, thus with zero boundary, in $U := \mathrm{R}\times \mathbb {R}^2$, decomposable in its regular part $T_r = [[G_{\mathbf{u}}]]$ and its singular part $T_s=\tau (S_s, \theta , \zeta )$; recall that $\mathcal L^2(p(S_s))=0$. By the lower semicontinuity of the mass we have

$$\begin{aligned} {\mathbf {M}}(T)\le \liminf _{h\rightarrow +\infty } {\mathbf {M}}([[G_{{\mathbf{u}}_h}]])=\overline{{\mathcal {A}}}({\mathbf{u}},\mathrm{R}) \end{aligned}$$

and

$$\begin{aligned} {\mathbf {M}}(T)={\mathbf {M}}(T_r)+{\mathbf {M}}(T_s)={\mathbf {M}}([[G_{\mathbf{u}}]]) + {\mathbf {M}}(T_s) = \int _{\mathrm{R}} |{\mathcal {M}}(\nabla {\mathbf{u}})|\,\text{ d }t\,\text{ d }s+ {\mathbf {M}}(T_s). \end{aligned}$$

In addition, the support of $T_s$ is bounded in U, since $({\mathbf{u}}_h)$ is bounded in $L^\infty (\mathrm{R}; \mathbb {R}^2)$, and $\partial T_s = - \partial T_r = - \partial [[G_{\mathbf{u}}]] =[[\{(t,0,f(t),1)\}_{t\in (a,b)}]]-[[\{(t,0,f(t),0)\}_{t\in (a,b)}]]$ in U. Therefore $T_s$ satisfies all hypotheses of Proposition 8.6, hence f has to be constant, providing a contradiction. $\square$