Beyond the Classical Cauchy–Born Rule

Abstract

Physically motivated variational problems involving non-convex energies are often formulated in a discrete setting and contain boundary conditions. The long-range interactions in such problems, combined with constraints imposed by lattice discreteness, can give rise to the phenomenon of geometric frustration even in a one-dimensional setting. While non-convexity entails the formation of microstructures, incompatibility between interactions operating at different scales can produce nontrivial mixing effects which are exacerbated in the case of incommensurability between the optimal microstructures and the scale of the underlying lattice. Unraveling the intricacies of the underlying interplay between non-convexity, non-locality and discreteness represents the main goal of this study. While in general one cannot expect that ground states in such problems possess global properties, such as periodicity, in some cases the appropriately defined ‘global’ solutions exist, and are sufficient to describe the corresponding continuum (homogenized) limits. We interpret those cases as complying with a Generalized Cauchy–Born (GCB) rule, and present a new class of problems with geometrical frustration which comply with GCB rule in one range of (loading) parameters while being strictly outside this class in a complimentary range. A general approach to problems with such ‘mixed’ behavior is developed.

Appendices

Appendix: Variations of Boundary Data

In this appendix we state and prove some technical results which allow the modification of boundary values of test functions for the minimum problems used in various characterization of $Q_{\textbf{m}} f$. In particular, these results allow to assume that test functions be constant close to the endpoints of the domain.

Let ${\textbf{m}}=\{m_n\}_n$ be such that $m_n\ge 0$ for any n, and there exists ${\overline{n}}$ such that $m_n$ is not increasing for $n\ge {\overline{n}}$. Moreover, we assume the decay condition $m_n=o(n^{-\beta })_{n\rightarrow +\infty }$ for some $\beta >2$.

Let $F_\varepsilon$ be defined as in (2.8); that is,

$$\begin{aligned} F_\varepsilon (u;I)=\sum _{\varepsilon i,\varepsilon (i-1)\in I}\varepsilon \, f\Big (\frac{u_{i}-u_{i-1}}{\varepsilon }\Big )+ \sum _{\varepsilon i,\varepsilon j\in I}\varepsilon \, m_{|i-j|} \Big (\frac{u_{i}-u_{j}}{\varepsilon }\Big )^2 \end{aligned}$$

for I interval and $u\in {\mathcal {A}}_\varepsilon (I)$.

Lemma A.1

Let $L>0$ and $N_\varepsilon =\lfloor \frac{L}{\varepsilon }\rfloor$. Let $\alpha \in (\frac{2}{\beta },1)$. Assume that $u\in L^2(0,L)$ and $u^\varepsilon \in {\mathcal {A}}_\varepsilon ={\mathcal {A}}_\varepsilon (0,L)$ be such that (the piecewise-affine extension of) the sequence $u^\varepsilon$ converges to u in $L^2(0,L)$, and $\sup _\varepsilon (F_\varepsilon (u^\varepsilon ;[0,L])+\Vert u^\varepsilon \Vert ^2_{L^2})=S<+\infty$. Then, there exists ${\hat{u}}^{\varepsilon }\in {\mathcal {A}}_\varepsilon$ converging to u such that

• (i) ${\hat{u}}^\varepsilon _i={\hat{u}}^\varepsilon _0$ for $i\leqq \varepsilon ^{-\alpha }$, ${\hat{u}}^\varepsilon _i={\hat{u}}^\varepsilon _{N_\varepsilon }$ for $i\ge N_\varepsilon -\varepsilon ^{-\alpha }$;

• (ii) $F_\varepsilon ({\hat{u}}^\varepsilon ;[0,L])\leqq F_\varepsilon (u^\varepsilon ;[0,L])+r(\varepsilon )$, where the remainder r depends only on S and f(0), and $r(\varepsilon )\rightarrow 0$ as $\varepsilon \rightarrow 0$.

Proof

We choose $\alpha ^\prime \in (0,1-\alpha )$ and define $\lambda _\varepsilon =\varepsilon ^{\alpha ^\prime }$ and $M_\varepsilon =\lfloor \varepsilon ^{\alpha +\alpha ^\prime -1}\rfloor -1$. For $\varepsilon$ small enough we divide $(0,\lambda _\varepsilon ]$ and $[L-\lambda _\varepsilon ,L)$ in $M_\varepsilon +1$ intervals by setting

$$\begin{aligned} I^k_\varepsilon= & {} \Big (\frac{k\lambda _\varepsilon }{M_\varepsilon +1},\frac{(k+1)\lambda _\varepsilon }{M_\varepsilon +1}\Big ],\\ J^k_\varepsilon= & {} \Big [L-\frac{(k+1)\lambda _\varepsilon }{M_\varepsilon +1},L-\frac{k\lambda _\varepsilon }{M_\varepsilon +1}\Big ), \ \ k\in \{0,\dots , M_\varepsilon \}. \end{aligned}$$

Since

$$\begin{aligned} \frac{1}{\varepsilon }\sum _{k=1}^{M_\varepsilon } \sum _{\varepsilon i\in I_\varepsilon ^k, \varepsilon j\in I_\varepsilon ^{k-1}} m_{|i-j|}(u^\varepsilon _i-u^\varepsilon _j)^2\leqq F_\varepsilon (u^\varepsilon ;[0,L])\leqq S, \end{aligned}$$

then there exists $k_\varepsilon ^-\in \{1,\dots , M_\varepsilon \}$ such that

$$\begin{aligned} \frac{1}{\varepsilon } \sum _{\varepsilon i\in I_\varepsilon ^{k_\varepsilon ^-}, \varepsilon j\in I_\varepsilon ^{k_\varepsilon ^--1}} m_{|i-j|}(u^\varepsilon _i-u^\varepsilon _j)^2 \leqq \frac{S}{M_\varepsilon }. \end{aligned}$$

(A.1)

The same argument allows to find $k_\varepsilon ^+\in \{1,\dots , M_\varepsilon \}$ such that the same inequality holds for $\varepsilon i\in J_\varepsilon ^{k_\varepsilon ^+}, \varepsilon j\in J_\varepsilon ^{k_\varepsilon ^+-1}$. Setting $j^-_\varepsilon =\min \{j: \varepsilon j\in I_\varepsilon ^{k_\varepsilon ^-}\}$ and $j^+_\varepsilon =\max \{j: \varepsilon j\in J_\varepsilon ^{k_\varepsilon ^+}\}$, we define ${\hat{u}}^{\varepsilon }$ by setting

$$\begin{aligned} {\hat{u}}^{\varepsilon }_i=\left\{ \begin{array}{ll} u^\varepsilon _{j^-_\varepsilon } &{} \hbox { if } i\leqq j^-_\varepsilon \\ u^\varepsilon _i &{} \hbox { if } j^-_\varepsilon \leqq i\leqq j^+_\varepsilon \\ u^\varepsilon _{j^+_\varepsilon } &{} \hbox { if } i\ge j^+_\varepsilon . \end{array} \right. \end{aligned}$$

(A.2)

Since $j_\varepsilon ^{-}\ge L\varepsilon ^{-\alpha }$ and $j_\varepsilon ^{+}\leqq N_\varepsilon -L\varepsilon ^{-\alpha }$, then ${\hat{u}}^\varepsilon$ satisfies claim (i). Moreover, ${\hat{u}}^\varepsilon \rightarrow u$ as $\varepsilon \rightarrow 0$. To prove this, for simplicity we suppose that $m_n$ is not increasing for $n\ge 1$. Then,

$$\begin{aligned} \varepsilon \sum _{i=1}^{j^-_\varepsilon }(u^\varepsilon _i-{\hat{u}}^\varepsilon _{i})^2= & {} \varepsilon \sum _{i=1}^{j^-_\varepsilon }(u^\varepsilon _i-u^\varepsilon _{j^-_\varepsilon })^2\leqq \varepsilon \sum _{i=1}^{j^-_\varepsilon } j^-_\varepsilon \!\!\sum _{j=i+1}^{j^-_\varepsilon }\!(u^\varepsilon _i-u^\varepsilon _{i-1})^2\\\leqq & {} \frac{S}{m_1}\varepsilon ^2 (j^-_\varepsilon )^2\leqq \frac{S}{m_1}\lambda _\varepsilon ^2, \end{aligned}$$

and correspondingly $\varepsilon \sum _{i=j^+_\varepsilon }^{\lfloor L/\varepsilon \rfloor }(u^\varepsilon _i-{\hat{u}}^\varepsilon _{i})^2\leqq \frac{S}{m_1}\lambda _\varepsilon ^2$. Setting, $n_\varepsilon =\lfloor \frac{\lambda _\varepsilon }{\varepsilon (M_\varepsilon +1)}\rfloor$, since

$$\begin{aligned} \sum _{|i-j|\ge n_\varepsilon } m_{|i-j|}(u^{\varepsilon }_i-u^{\varepsilon }_j)^2 \leqq \frac{2}{\varepsilon } m_{n_\varepsilon } \Vert u^\varepsilon \Vert ^2_{L_2}\leqq \frac{2}{\varepsilon } m_{\lfloor \varepsilon ^{-\alpha }\rfloor } \Vert u^\varepsilon \Vert ^2_{L_2}, \end{aligned}$$

and recalling (A.1), we obtain

$$\begin{aligned} F_\varepsilon ({\hat{u}}^{\varepsilon };[0,L])\leqq F_\varepsilon (u^{\varepsilon };[0,L])+ 2\lambda _\varepsilon f(0) + \frac{C}{\varepsilon ^2} m_{\lfloor \varepsilon ^{-\alpha }\rfloor }+\frac{C}{M_\varepsilon }, \end{aligned}$$

where C denotes a constant depending only on $\sup _\varepsilon F_\varepsilon (u^\varepsilon ;[0,L])$ and $\sup _\varepsilon \Vert u^\varepsilon \Vert _{L^2}$. Setting

$$\begin{aligned} r(t)=2f(0) t^{\alpha ^\prime }+Ct^{\alpha \beta -2}+C t^{1-\alpha -\alpha ^\prime }, \end{aligned}$$

we conclude the proof since $m_n=o(n^{-\beta })$ and $\alpha >\frac{2}{\beta }$. $\quad \square$

Let $a,b>0$. We define the functional $E_\varepsilon (u,v;I)$ by setting

$$\begin{aligned} E_\varepsilon (u,v;I)= & {} \sum _{\varepsilon i,\varepsilon (i-i)\in I}\varepsilon \, f\Big (\frac{u_i-u_{i-1}}{\varepsilon }\Big )\nonumber \\{} & {} + \frac{a}{2}\sum _{\varepsilon i,\varepsilon (i-i)\in I}\varepsilon \, \Big (\frac{v_{i}-v_{i-1}}{\varepsilon }\Big )^2+\frac{b}{2\varepsilon }\sum _{\varepsilon i\in I}(u_i-v_i)^2 \end{aligned}$$

(A.3)

for I interval and $u,v\in {\mathcal {A}}_\varepsilon (I)$.

Lemma A.2

Let $L>0$ and $N_\varepsilon =\lfloor \frac{L}{\varepsilon }\rfloor$. Let $\alpha \in (\frac{2}{\beta },1)$. Assume that $u^\varepsilon , v^\varepsilon \in {\mathcal {A}}_\varepsilon$ be such that (the piecewise-affine extensions of) $u^\varepsilon$ and $v^\varepsilon$ converge to u in $L^2(0,L)$ and $\sup _\varepsilon (E_\varepsilon (u^\varepsilon ;[0,L])+\Vert u^\varepsilon \Vert ^2_{L^2})=S<+\infty$. Then there exist ${\hat{u}}^{\varepsilon }, {\hat{v}}^\varepsilon \in {\mathcal {A}}_\varepsilon$ converging to u such that

• (i) ${\hat{u}}^\varepsilon _i={\hat{v}}^\varepsilon _i={\hat{u}}^\varepsilon _0$ for $i\leqq \varepsilon ^{-\alpha }$, ${\hat{u}}^\varepsilon _i={\hat{v}}^\varepsilon _i={\hat{u}}^\varepsilon _{N_\varepsilon }$ for $i\ge N_\varepsilon -\varepsilon ^{-\alpha }$;

• (ii) $E_\varepsilon ({\hat{u}}^\varepsilon , {\hat{v}}^\varepsilon ;[0,L])\leqq E_\varepsilon (u^\varepsilon , v^\varepsilon ;[0,L])+r(\varepsilon )$, where the remainder r depends only on S and f(0), and $r(\varepsilon )\rightarrow 0$ as $\varepsilon \rightarrow 0$.

Proof

We choose $\lambda _\varepsilon$ and $M_\varepsilon$ as in the proof of Lemma A.1, and divide $(0,\lambda _\varepsilon ]$ and $[L-\lambda _\varepsilon ,L)$ in $M_\varepsilon +1$ intervals, denoted by $I^k_\varepsilon$ and $J^k_\varepsilon$ respectively, as above. Then, there exist $k_\varepsilon$ and $h_\varepsilon$ in $\{1,\dots , M_\varepsilon \}$ such that

$$\begin{aligned} \frac{1}{2\varepsilon } \sum _{\varepsilon i\in I_\varepsilon ^{k_\varepsilon }\cup J_\varepsilon ^{h_\varepsilon }} \big (a(v^\varepsilon _i-v^\varepsilon _{i-1})^2+b(u^\varepsilon _i-v^\varepsilon _{i})^2 \big ) \leqq \frac{S}{M_\varepsilon }. \end{aligned}$$

(A.4)

Setting $j^-_\varepsilon =\min \{j: \varepsilon j\in I_\varepsilon ^{k_\varepsilon }\}$ and $j^+_\varepsilon =\max \{j: \varepsilon j\in J_\varepsilon ^{h_\varepsilon }\}$, we define

$$\begin{aligned} {\hat{u}}^{\varepsilon }_i=\left\{ \begin{array}{ll} u^\varepsilon _{j_{\varepsilon }^-} &{} \hbox { if } i\leqq j_{\varepsilon }^- \\ u^\varepsilon _i &{} \hbox { if } j_{\varepsilon }^-< i< j_{\varepsilon }^+ \\ u^\varepsilon _{j_{\varepsilon }^+} &{} \hbox { if } i\ge j_{\varepsilon }^+ \end{array} \right. \ \ \hbox { and } \ \ \ \ {\hat{v}}^{\varepsilon }_i=\left\{ \begin{array}{ll} u^\varepsilon _{j_{\varepsilon }^-} &{} \hbox { if } i\leqq j_{\varepsilon }^- \\ v^\varepsilon _i &{} \hbox { if } j_{\varepsilon }^-< i < j_{\varepsilon }^+ \\ u^\varepsilon _{j_{\varepsilon }^+} &{} \hbox { if } i\ge j_{\varepsilon }^+, \end{array} \right. \end{aligned}$$

so that ${\hat{u}}^\varepsilon$ and ${\hat{v}}^\varepsilon$ converge to u in $L^2$, and satisfy (i). Recalling (A.4), we get in particular that

$$\begin{aligned} \frac{a}{2\varepsilon } ({\hat{v}}^\varepsilon _{j_{\varepsilon }^-+1}-{\hat{v}}^\varepsilon _{j_{\varepsilon }^-})^2\leqq \frac{a}{\varepsilon } (v^\varepsilon _{j_{\varepsilon }^-+1}-v^\varepsilon _{j_{\varepsilon }^-})^2+ \frac{a}{\varepsilon } (v^\varepsilon _{j_{\varepsilon }^-}-u^\varepsilon _{j_{\varepsilon }^-})^2 \leqq \frac{C}{M_\varepsilon }, \end{aligned}$$

where C denotes a positive constant depending only on a, b and S. The same bound holds for $\frac{a}{2\varepsilon }({\hat{v}}^\varepsilon _{j_{\varepsilon }^+}-{\hat{v}}^\varepsilon _{j_{\varepsilon }^+-1})^2$. Hence

$$\begin{aligned} E_\varepsilon ({\hat{u}}_\varepsilon ,{\hat{v}}_\varepsilon ; [0,L])\leqq & {} 2\lambda _\varepsilon f(0) +E_\varepsilon (u^\varepsilon ,v^\varepsilon ;(0,L)) \\{} & {} +\frac{a}{2\varepsilon } ({\hat{v}}^\varepsilon _{j_{\varepsilon }^-+1}-{\hat{v}}^\varepsilon _{j_{\varepsilon }^-})^2 + \frac{a}{2\varepsilon }({\hat{v}}^\varepsilon _{j_{\varepsilon }^+}-{\hat{v}}^\varepsilon _{j_{\varepsilon }^+-1})^2 \\\leqq & {} 2\lambda _\varepsilon f(0) +E_\varepsilon (u^\varepsilon ,v^\varepsilon ;[0,L]) + \frac{2C}{M_\varepsilon }, \end{aligned}$$

concluding the proof as above. $\quad \square$

Remark A.3

In the hypotheses of Lemma A.2, if there exists $\alpha \in (0,1)$ such that $u^\varepsilon _i={\hat{u}}^\varepsilon _0$ for $i\leqq \varepsilon ^{-\alpha }$ and $u^\varepsilon _i={\hat{u}}^\varepsilon _{N_\varepsilon }$ for $i\ge N_\varepsilon -\varepsilon ^{-\alpha }$ for some $\alpha >0$, then the function ${\hat{v}}^\varepsilon$ can be chosen such that it coincides with $u^\varepsilon$ for $i\leqq \varepsilon ^{-\alpha ^{\prime \prime }}$ and for $i\ge N_\varepsilon -\varepsilon ^{-\alpha ^{\prime \prime }}$ with $\alpha ^{\prime \prime }<\alpha$.

Appendix: Formulas for $P^{M,n}$ in the Concentrated Case

In this appendix we include some explicit computations of the functions $P^{M,n}$ defined in (3.1), which are the energies of the locking states $n\over M$ in the concentrated case. The formulas of these functions have been used in Sections 4.2.1 and 4.2.2 to highlight the structure of $Q_{\textbf{m}}f(z)$ in the truncated-parabolic and double-well case, respectively. Here, we include the corresponding computations.

Truncated-parabolic case. Let f be given by (4.10). In view of (4.3), the domains of $P^{M,0}$ and $P^{M,M}$ are $\{z\leqq 1\}$ and $\{z\ge 1\}$, respectively. We recall that here

$$\begin{aligned} P^{M,0}(z)=z^2+2(m_1+m_M M^2)z^2 \ \ {\textrm{and }}\ \ P^{M,M}(z)=1+2(m_1+m_M M^2)z^2. \end{aligned}$$

For $n=1,\dots , M-1$, we can also write

$$\begin{aligned} P^{M,n}(z)={\left\{ \begin{array}{ll} \displaystyle \frac{2m_1+1}{1-\theta _n}\big (z^2-\theta _n(2z-1)\big )+2m_M M^2z^2 &{} \displaystyle {\textrm{if }}\ z\leqq T_n^{-}\\ \displaystyle \theta _n+\frac{2m_1(2m_1+1)}{2m_1+\theta _n}z^2 +2m_M M^2 z^2 &{} \displaystyle {\textrm{if }}\ T_n^{-}\leqq z \leqq T_n^{+}\\ \displaystyle 1+\frac{2m_1}{\theta _n}\big ((z-1)^2+\theta _n(2z-1)\big ) +2m_M M^2 z^2 &{} \displaystyle {\textrm{if }}\ z\ge T_n^{+}, \end{array}\right. }\nonumber \\ \end{aligned}$$

(B.1)

where

$$\begin{aligned} T_n^{-}=\frac{2m_1+\theta _n}{2m_1+1} \ \ {\textrm{and }}\ \ T_n^{+}=\frac{2m_1+\theta _n}{2m_1}. \end{aligned}$$

Fig. 32

Envelope of two consecutive functions $P^{M,n}(z)$

Note that while the formula defining $P^{M,n}$ changes form at $z=T_n^{-}$ and $z=T_n^{+}$, the computation of the common tangent points of $P^{M,n}$ and $P^{M,n+1}$ involves only the central formula in (B.1). Consequently, the points $s_n^+$ and $s_n^-$ in Theorem 4.1 are

$$\begin{aligned} \begin{array}{ll} &{}\displaystyle s_n^+=s_n^+(m_1,m_M) =\frac{2m_1+\theta _n}{\sqrt{2m_1(2m_1+1)}}\ \sqrt{\frac{m_1(2m_1+1)+m_M M^2(2m_1+\theta _{n+1})}{m_1(2m_1+1)+m_M M^2(2m_1+\theta _{n})}} \\ &{}\displaystyle s_n^-=s_n^-(m_1,m_M) = \frac{2m_1+\theta _n}{\sqrt{2m_1(2m_1+1)}}\ \sqrt{\frac{m_1(2m_1+1)+m_M M^2(2m_1+\theta _{n-1})}{m_1(2m_1+1)+m_M M^2(2m_1+\theta _{n})}}. \end{array}\nonumber \\ \end{aligned}$$

(B.2)

In Fig. 32 we illustrate the envelope of two consecutive functions $P^{M,n}(z)$, bridging energies of consecutive locking states with an affine function.

Finally, since $s_n^+\ge T_n^-$ and $s_n^-\leqq T_n^+$, we have the formula

$$\begin{aligned} Q_{\textbf{m}}f(z)={\left\{ \begin{array}{ll} \displaystyle z^2&{} {\textrm{if }} \ z\leqq s^+_0\\ \displaystyle r^{M,n}(z)-2(m_1+m_M M^2)z^2 &{} {\textrm{if }} \ s_n^+ \leqq z \leqq s_{n+1}^- \\ \displaystyle \frac{2m_1(1-\theta _n)}{2m_1+\theta _n}z^2+\theta _n&{} {\textrm{if }} \ s_n^- \leqq z \leqq s_{n}^+ \\ \displaystyle 1 &{} {\textrm{if }} \ s_M^- \leqq z, \end{array}\right. } \end{aligned}$$

(B.3)

where $r^{M,n}$ is the affine function

$$\begin{aligned} r^{M,n}(z)=P^{M,n}(s_n^+)+\frac{2(z-s_n^+)}{M(s_{n+1}^--s_n^+)}. \end{aligned}$$

Bi-quadratic double-well case. Let f be given by $f(z)=(1-|z|)^2$. By using (4.3) the domains of $P^{M,0}$ and $P^{M,M}$ are $\{z\leqq 0\}$ and $\{z\ge 0\}$, respectively, where

$$\begin{aligned} P^{M,0}(z)= & {} (1+z)^2+ 2(m_1+m_MM^2)z^2 \ \ {\textrm{and }} \\ P^{M,M}(z)= & {} (1-z)^2+ 2(m_1+m_MM^2)z^2. \end{aligned}$$

For $n=1,\dots , M-1$

$$\begin{aligned} P^{M,n}(z)={\left\{ \begin{array}{ll} \displaystyle \Big (\frac{1+2m_1}{1-\theta _n}+2m_M M^2\Big )z^2+2z+1 &{} \displaystyle {\textrm{if }}\ z\leqq T_n^-\\ \displaystyle (1+z)^2 +\displaystyle 2(m_1+m_M M^2)z^2-4\theta _n\Big (z+\frac{1-\theta _n}{1+2m_1}\Big )&{} \displaystyle {\textrm{if }}\ T_n^-\leqq z\leqq T_n^+\\ \displaystyle \Big (\frac{1+2m_1}{\theta _n}+2m_M M^2\Big )z^2-2z+1 &{} \displaystyle {\textrm{if }}\ z\ge T_n^+, \end{array}\right. } \end{aligned}$$

where in this case the points $T_n^-$ and $T_n^+$ where the formula changes are given by

$$\begin{aligned} T_n^-=-\frac{2(1-\theta _n)}{1+2m_1} \ \ {\textrm{and }}\ \ T_n^+=\frac{2\theta _n}{1+2m_1}. \end{aligned}$$

Consequently,

$$\begin{aligned} \left. \begin{array}{ll} &{}\displaystyle s_n^+(m_1,m_M)=s_n^+= \frac{2m_M M}{(1+2m_1)(1+2m_1+2m_M M^2)}+\frac{2\theta _n-1}{1+2m_1}\\ &{}\displaystyle s_n^-(m_1,m_M)=s_n^-=-\frac{2m_M M}{(1+2m_1)(1+2m_1+2m_M M^2)}+\frac{2\theta _n-1}{1+2m_1}. \end{array} \right. \end{aligned}$$

(B.4)

Since $s_n^+\ge T_n^-$ and $s_n^-\leqq T_n^+$, we obtain

$$\begin{aligned} Q_{\textbf{m}}f(z)={\left\{ \begin{array}{ll} (1+z)^2&{} {\textrm{if }} \ z\leqq s_0^+\\ r^{M,n}(z)-2(m_1+m_M M^2) z^2 &{} {\textrm{if }} \ s_n^+ \leqq z \leqq s_{n+1}^- \\ \displaystyle z^2+2(1-2\theta _n)z+1-\frac{4\theta _n(1-\theta _n)}{1+2m_1}&{} {\textrm{if }} \ s_n^- \leqq z \leqq s_{n}^+ \\ (1-z)^2 &{} {\textrm{if }} \ s_M^- \leqq z, \end{array}\right. } \end{aligned}$$

where $r^{M,n}$ is the affine function

$$\begin{aligned} r^{M,n}(z)= & {} P^{M,n}(s_n^+)+\frac{M(1+2(m_1+m_M M^2))}{2}\\{} & {} \big (P^{M,n+1}(s_{n+1}^-)-P^{M,n}(s_n^+)\big )(z-s_n^+). \end{aligned}$$