Quantitative immersability of Riemann metrics and the infinite hierarchy of prestrained shell models

TL;DR

This paper analyzes the asymptotic behavior of prestrained thin shells by studying the energy deficits of their immersions, deriving a hierarchy of models based on Riemannian curvature conditions and energy scalings.

Contribution

It completes the scaling analysis of non-Euclidean energies for thin shells, identifying all possible energy regimes and their geometric conditions, extending previous results.

Findings

Energy quantization occurs at even powers of thickness h.

Scaling regimes are characterized by vanishing Riemann curvatures.

Asymptotic behavior of minimizing immersions is established.

Abstract

This paper concerns the variational description of prestrained materials, in the context of dimension reduction for thin films $\Omega^h=\omega\times (-\frac{h}{2}, \frac{h}{2})$. Given a Riemann metric $G$ on $\Omega^1$, we study the question of what is the infimum of the averaged pointwise deficit of a given immersion from being an orientation-preserving isometric immersion of $G_{\mid \Omega^h}$ on $\Omega^h,$ over all weakly regular immersions. This deficit is measured by the non-Euclidean energies $\mathcal{E}^h$, which can be seen as modifications of the classical nonlinear three-dimensional elasticity. Building on our previous results, we complete the scaling analysis of $\mathcal{E}^h$ and the derivation of $\Gamma$-limits of the scaled energies $h^{-2n}\mathcal{E}^h$, for all $n\geq 1$. We show the energy quantisation in the sense that the even powers $2n$ of $h$ are indeed the only possible ones (all of them are also attained). For each $n$, we identify the equivalent conditions for the validity of the corresponding scaling, in terms of the vanishing of appropriate Riemann curvatures of $G$ to certain orders, and in terms of the matched isometry expansions. We also establish the asymptotic behaviour of the minimizing immersions as $h\to 0$.

Equations

$$\Omega^{h}=\omega\times\Big{(}-\frac{h}{2},\frac{h}{2}\Big{)}\subset\mathbb{R}^{3},\qquad 0<h\ll 1.$$

$$G\in {\mathcal{C}}^{\infty }({\bar{\Omega }}^1,{\mathbb{R}}_{\mathrm{sym},\mathrm{pos}}^{3\times 3})$$

$$\mathcal{E}^{h}(u^{h})=\frac{1}{h}\int_{\Omega^{h}}W\big{(}\nabla u^{h}(x)G(x)^{-1/2}\big{)}{\;\rm d}x=\int_{\Omega^{1}}W\big{(}\nabla u^{h}(x^{\prime},hx_{3})G(x^{\prime},hx_{3})^{-1/2}\big{)}{\;\rm d}x,$$

$$\mathbb{E}\doteq\big{(}L^{2}(\omega,\mathbb{R}^{2\times 2}_{{\rm sym}\,}),\|\cdot\|_{\mathcal{Q}_{2}}\big{)},\qquad\|F\|_{\mathcal{Q}_{2}}=\Big{(}\int_{\omega}\mathcal{Q}_{2}(x^{\prime},F(x^{\prime})){\;\rm d}x^{\prime}\Big{)}^{1/2}.$$

$$\begin{split}&\mathcal{Q}_{3}(F)=D^{2}W(Id_{3})(F,F)\\ &\mathcal{Q}_{2}(x^{\prime},F_{2\times 2})=\min\left\{\mathcal{Q}_{3}\big{(}G(x^{\prime},0)^{-1/2}\tilde{F}G(x^{\prime},0)^{-1/2}\big{)};\leavevmode\nobreak\ \tilde{F}\in\mathbb{R}^{3\times 3}\mbox{ with }\tilde{F}_{2\times 2}=F_{2\times 2}\right\}.\end{split}$$

$$F_{2\times 2}\mapsto c(x^{\prime},F_{2\times 2})\in\mathbb{R}^{3}\;\;\mbox{ with:}\quad\mathcal{Q}_{2}(x^{\prime},F_{2\times 2})=\mathcal{Q}_{3}\Big{(}G(x^{\prime},0)^{-1/2}\big{(}F_{2\times 2}^{*}+c\otimes e_{3}\big{)}G(x^{\prime},0)^{-1/2}\Big{)}.$$

$$\underset{h\to 0}{\lim }\frac{1}{h^4}\inf {\mathcal{E}}^h=0\iff R_{ab,cd}(x^{\prime },0)=0\text{mbox}forall{x}^{\prime }\in \omega ,\text{leavevmode}\text{leavevmode}\text{mbox}foralla,b,c,d:1\dots 3.$$

$$B_{0}=\big{[}\partial_{1}y_{0},\leavevmode\nobreak\ \partial_{2}y_{0},\leavevmode\nobreak\ \vec{b}_{1}\big{]},\qquad B_{1}=\big{[}\partial_{1}\vec{b}_{1},\leavevmode\nobreak\ \partial_{2}\vec{b}_{1},\leavevmode\nobreak\ \vec{b}_{2}\big{]},$$

$$\begin{split}&B_{0}^{\scriptscriptstyle{\mathsf{T}}}B_{0}=G(x^{\prime},0)\quad\mbox{ with }\quad\det B_{0}>0\\ &\mbox{and }\quad(B_{0}^{\scriptscriptstyle{\mathsf{T}}}B_{1})_{{\rm sym}\,}=\frac{1}{2}\partial_{3}G(x^{\prime},0)\\ &\mbox{and }\quad\big{(}(\nabla y_{0})^{\scriptscriptstyle{\mathsf{T}}}\nabla\vec{b}_{2}\big{)}_{{\rm sym}\,}+(\nabla\vec{b}_{1})^{\scriptscriptstyle{\mathsf{T}}}\nabla\vec{b}_{1}=\frac{1}{2}\partial_{33}G(x^{\prime},0)_{2\times 2}.\end{split}$$

$$\Big{(}\sum_{k=0}^{2}\frac{x_{3}^{k}}{k!}B_{k}\Big{)}^{\scriptscriptstyle{\mathsf{T}}}\Big{(}\sum_{k=0}^{2}\frac{x_{3}^{k}}{k!}B_{k}\Big{)}=G(x^{\prime},x_{3})+\mathcal{O}(h^{3})\quad\mbox{ on }\Omega^{h},\quad\mbox{ as }h\to 0.$$

$$\Big{(}\sum_{k=0}^{n}\frac{x_{3}^{k}}{k!}B_{k}\Big{)}^{\scriptscriptstyle{\mathsf{T}}}\Big{(}\sum_{k=0}^{n}\frac{x_{3}^{k}}{k!}B_{k}\Big{)}=G(x^{\prime},x_{3})+\mathcal{O}(h^{n+1})\quad\mbox{ on }\Omega^{h},\quad\mbox{ as }h\to 0,$$

$$V^{h}(x^{\prime})=\frac{1}{h^{n}}\fint_{-h/2}^{h/2}(\bar{R}^{h})^{\scriptscriptstyle{\mathsf{T}}}\big{(}u^{h}(x^{\prime},x_{3})-c^{h}\big{)}-\Big{(}y_{0}(x^{\prime})+\sum_{k=1}^{n}\frac{x_{3}^{k}}{k!}\vec{b}_{k}(x^{\prime})\Big{)}{\;\rm d}x_{3}$$

$$V\in\mathcal{V}_{y_{0}}=\Big{\{}V\in W^{2,2}(\omega,\mathbb{R}^{3});\leavevmode\nobreak\ \big{(}(\nabla y_{0})^{\scriptscriptstyle{\mathsf{T}}}\nabla V\big{)}_{{\rm sym}\,}=0\quad\mbox{a.e. in }\omega\Big{\}}.$$

$$\big{(}B_{0}^{\scriptscriptstyle{\mathsf{T}}}\big{[}\nabla V,\leavevmode\nobreak\ \vec{p}\big{]}\big{)}_{{\rm sym}\,}=0\quad\mbox{ a.e. in }\omega,$$

$$\begin{split}\mathcal{I}_{2(n+1)}(V)=&\frac{1}{24}\cdot\Big{\|}(\nabla y_{0})^{\scriptscriptstyle{\mathsf{T}}}\nabla\vec{p}+(\nabla V)^{\scriptscriptstyle{\mathsf{T}}}\nabla\vec{b}_{1}+\alpha_{n}\big{[}\partial_{3}^{(n-1)}R_{i3,j3}\big{]}_{i,j=1\ldots 2}\Big{\|}^{2}_{\mathcal{Q}_{2}}\\ &+\beta_{n}\cdot\Big{\|}\mathbb{P}_{\mathcal{S}_{y_{0}}^{\perp}}\big{(}\big{[}\partial_{3}^{(n-1)}R_{i3,j3}\big{]}_{i,j=1\ldots 2}\big{)}\Big{\|}^{2}_{\mathcal{Q}_{2}}\\ &+\gamma_{n}\cdot\Big{\|}\mathbb{P}_{\mathcal{S}_{y_{0}}}\big{(}\big{[}\partial_{3}^{(n-1)}R_{i3,j3}\big{]}_{i,j=1\ldots 2}\big{)}\Big{\|}^{2}_{\mathcal{Q}_{2}}.\end{split}$$

$$\mathcal{S}_{y_{0}}=\mathrm{closure}_{\mathbb{E}}\Big{\{}\big{(}(\nabla y_{0})^{\scriptscriptstyle{\mathsf{T}}}\nabla w\big{)}_{{\rm sym}\,};\leavevmode\nobreak\ w\in W^{1,2}(\omega,\mathbb{R}^{3})\Big{\}},$$

$$\begin{split}&\alpha_{n}=\left\{\begin{array}[]{ll}0&\mbox{for }\;n\mbox{ odd}\\ \displaystyle\frac{3}{2^{n}(n+3)(n+1)!}&\mbox{for }\;n\mbox{ even}\end{array}\right.,\\ &\beta_{n}=\frac{1}{2^{2n+3}(2n+3)\big{(}(n+1)!\big{)}^{2}}\cdot\left\{\begin{array}[]{ll}1&\mbox{for }\;n\mbox{ odd}\\ \displaystyle\frac{n^{2}}{(n+3)^{2}}&\mbox{for }\;n\mbox{ even}\end{array}\right.,\\ &\gamma_{n}=\frac{1}{2^{2n+3}(2n+3)\big{(}(n+1)!\big{)}^{2}}\cdot\left\{\begin{array}[]{ll}\displaystyle\frac{(n+1)^{2}}{(n+2)^{2}}&\mbox{for }\;n\mbox{ odd}\\ \displaystyle\frac{n^{2}}{(n+3)^{2}}&\mbox{for }\;n\mbox{ even}\end{array}\right..\end{split}$$

$$\begin{split}2\cdot\big{[}\partial_{3}^{(n-1)}R_{i3,j3}(\cdot,0)\big{]}_{i,j=1\ldots 2}=\;&2\big{(}(\nabla y_{0})^{\scriptscriptstyle{\mathsf{T}}}\nabla\vec{b}_{n+1}\big{)}_{{\rm sym}\,}+\sum_{k=1}^{n}\binom{n+1}{k}(\nabla\vec{b}_{k})^{\scriptscriptstyle{\mathsf{T}}}\nabla\vec{b}_{n+1-k}\\ &-\partial_{3}^{(n+1)}G(\cdot,0)_{2\times 2}.\end{split}$$

$$\frac{1}{h^{n}}\fint_{-h/2}^{h/2}u^{h}(x^{\prime},x_{3})-\Big{(}y_{0}+\sum_{k=1}^{n}\frac{x_{3}^{k}}{k!}\vec{b}_{k}\Big{)}{\;\rm d}x_{3}\to V\quad\mbox{as }h\to 0,\quad\mbox{strongly in }\;W^{1,2}(\omega,\mathbb{R}^{3}),$$

$$c{h}^{2(n+1)}\le \inf {\mathcal{E}}^h\le C{h}^{2(n+1)}.$$

$$\inf {\mathcal{E}}^h\le C{h}^{2(n+1)}.$$

$$\nabla u^{h}(x^{\prime},x_{3})=\sum_{k=0}^{n}\frac{x_{3}^{k}}{k!}B_{k}+\frac{x_{3}^{n+1}}{(n+1)!}\big{[}\partial_{1}\vec{b}_{n+1},\leavevmode\nobreak\ \partial_{2}\vec{b}_{n+1},\leavevmode\nobreak\ 0\big{]}.$$

$$\begin{split}\sqrt{\big{(}(\nabla u^{h})G^{-1/2}\big{)}^{\scriptscriptstyle{\mathsf{T}}}\big{(}(\nabla u^{h})G^{-1/2}\big{)}}&=\sqrt{Id_{3}+G^{-1/2}\big{(}(\nabla u^{h})^{\scriptscriptstyle{\mathsf{T}}}\nabla u^{h}-G\big{)}G^{-1/2}}\\ &=\sqrt{Id_{3}+\mathcal{O}(h^{n+1})}=Id_{3}+\mathcal{O}(h^{n+1}).\end{split}$$

$$\frac{1}{h}\int_{V^{h}}\big{|}\nabla u^{h}-\bar{R}^{h}\sum_{k=0}^{n}\frac{x_{3}^{k}}{k!}B_{k}\big{|}^{2}{\;\rm d}x\leq C\Big{(}\frac{1}{h}\int_{V^{h}}W\big{(}(\nabla u^{h})G^{-1/2}\big{)}{\;\rm d}x+h^{2n+1}|V^{h}|\Big{)}.$$

$$\int_{U^{h}}|\nabla v^{h}-\bar{R}^{h}|^{2}\leq C\int_{U^{h}}{\rm dist}^{2}\big{(}\nabla v^{h},SO(3)\big{)}.$$

$$\begin{split}\int_{U^{h}}|\nabla v^{h}-\bar{R}^{h}|^{2}&=\int_{V^{h}}(\det\nabla Y)\cdot\big{|}(\nabla u^{h})(\nabla Y)^{-1}-\bar{R}^{h}\big{|}^{2}\geq C\int_{V^{h}}\big{|}\nabla u^{h}-\bar{R}^{h}\nabla Y\big{|}^{2}\\ &=C\int_{V^{h}}\big{|}\nabla u^{h}-\bar{R}^{h}\big{(}\sum_{k=0}^{n}\frac{x_{3}^{k}}{k!}B_{k}\big{)}\big{|}^{2}+C\int_{V^{h}}\mathcal{O}(h^{2(n+1)}).\end{split}$$

$$\begin{split}\int_{U^{h}}{\rm dist}^{2}\big{(}\nabla v^{h},SO(3)\big{)}&=\int_{V^{h}}(\det\nabla Y)\cdot{\rm dist}^{2}\big{(}(\nabla u^{h})(\nabla Y)^{-1},SO(3)\big{)}\\ &\leq C\int_{V^{h}}{\rm dist}^{2}\big{(}(\nabla v^{h})G^{-1/2},SO(3)(\nabla Y)G^{-1/2}\big{)}\\ &\leq C\int_{V^{h}}{\rm dist}^{2}\big{(}(\nabla v^{h})G^{-1/2},SO(3)\big{)}+C\int_{V^{h}}\mathcal{O}(h^{2(n+1)}).\end{split}$$

$$\begin{split}\frac{1}{h}\int_{\Omega^{h}}\big{|}\nabla u^{h}-R^{h}\sum_{k=0}^{n}\frac{x_{3}^{k}}{k!}B_{k}\big{|}^{2}{\;\rm d}x\leq Ch^{2(n+1)}\quad\mbox{ and }\quad\int_{\omega}|\nabla R^{h}(x^{\prime})|^{2}{\;\rm d}x^{\prime}\leq Ch^{2n}.\end{split}$$

$$\begin{split}{\rm dist}^{2}\Big{(}\fint_{\Omega^{h}}(\nabla u^{h})&\Big{(}\sum_{k=0}^{n}\frac{x_{3}^{k}}{k!}B_{k}\Big{)}^{-1}{\;\rm d}x,SO(3)\Big{)}\leq\Big{|}\fint_{\Omega^{h}}(\nabla u^{h})\Big{(}\sum_{k=0}^{n}\frac{x_{3}^{k}}{k!}B_{k}\Big{)}^{-1}{\;\rm d}x-R^{h}(x^{\prime})\Big{|}^{2}\\ &\leq 2\fint_{\Omega^{h}}\big{|}(\nabla u^{h})\Big{(}\sum_{k=0}^{n}\frac{x_{3}^{k}}{k!}B_{k}\Big{)}^{-1}-R^{h}\big{|}^{2}{\;\rm d}x+2\big{|}\big{(}\fint_{\omega}R^{h}{\;\rm d}x^{\prime}\big{)}-R^{h}(x^{\prime})\big{|}^{2},\end{split}$$

$${\rm dist}^{2}\Big{(}\fint_{\Omega^{h}}(\nabla u^{h})\Big{(}\sum_{k=0}^{n}\frac{x_{3}^{k}}{k!}B_{k}\Big{)}^{-1}{\;\rm d}x,SO(3)\Big{)}\leq Ch^{2(n+1)}+Ch^{2n}\leq Ch^{2n}.$$

Full text

Building on our previous results, we complete the scaling analysis of $\mathcal{E}^{h}$ and the derivation of $\Gamma$-limits of the scaled energies $h^{-2n}\mathcal{E}^{h}$, for all $n\geq 1$. We show the energy quantisation, in the sense that the even powers $2n$ of $h$ are indeed the only possible ones (all of them are also attained). For each $n$, we identify the equivalent conditions for the validity of the corresponding scaling, in terms of the vanishing of appropriate Riemann curvatures of $G$ up to certain orders, and in terms of the matched isometry expansions. We also establish the asymptotic behaviour of the minimizing immersions as $h\to 0$.

1. Introduction

In this paper, we propose results that address and relate the following two contexts:

• (i)

Quantitative analysis of immersability of Riemann metrics.

• (ii)

Dimension reduction in non-Euclidean elasticity of prestrained thin films.

It is a well-known fact that a three-dimensional Riemann metric $G$ has a smooth isometric immersion in $\mathbb{R}^{3}$, if an only if its curvature tensor $R(G)=\{R_{ab,cd}\}_{a,b,c,d=1\ldots 3}$ vanishes identically. The smoothness requirement may be replaced by the orientation-preservation of a Lipschitz continuous immersion; then the condition $R(G)=0$ automatically yields smoothness and uniqueness, up to rigid motions. When $R(G)\neq 0$, one may pose the question of what is the infimum of the average pointwise deficit from being an orientation-preserving isometric immersion, over all, weakly regular, immersions. We study this question on a family of thin films \big{\{}\Omega^{h}=\omega\times(-\frac{h}{2},\frac{h}{2})\big{\}}_{h\to 0} around a given two-dimensional midplate $\omega$, where the said deficit is measured by the energy: $\mathcal{E}^{h}(u)=\fint_{\Omega^{h}}{\rm dist}^{2}((\nabla u)G^{-1/2},SO(3))$. Our first goal is to determine the possible scalings: $\inf\mathcal{E}^{h}\sim h^{\beta}$, as $h\to 0$, in terms of powers $\beta$ of the thickness $h$. We are then interested in identifying properties of $G$, that correspond to each scaling range, in function of the curvature components and their derivatives. Finally, we want to predict the asymptotics of the minimizing immersions as $h\to 0$.

Similar questions arise in the context of the so-called prestrained elasticity. A prestrained elastic body is a three-dimensional object, modeled in its reference configuration by a domain and a Riemann metric $G$, which is induced by mechanisms such as growth, plasticity, thermal expansion etc. The body wants to realize the distances between its constitutive cell elements, which are set by $G$, by deforming its shape. Since this realization is taking place in the flat three-dimensional space, it is impossible unless $R(G)=0$. This condition is precisely equivalent to having the stored non-Euclidean energy of deformations infimize to zero. In the variational description of thin prestrained films $\Omega^{h}$, we thus study the nonlinear energies: \big{\{}\mathcal{E}^{h}(u)=\fint_{\Omega^{h}}W((\nabla u)G^{-1/2})\}_{h\to 0} and, as above, want to determine the viable scalings of their infima, their singular limits as $h\to 0$, and the asymptotic behaviour of the three-dimensional minimizing shapes.

In our previous works [28, 6] we analyzed the scenario: $\inf\mathcal{E}^{h}\sim h^{2}$, whereas in [29, 30] we showed that the next limiting energy level beyond $h^{2}$ is: $\inf\mathcal{E}^{h}\sim h^{4}$, arising when $\{R_{12,ab}\}_{a,b=1\ldots 3}=0$ on $\omega$. Then we observed that the further scaling level is: $\inf\mathcal{E}^{h}\sim h^{6}$ and that it corresponds to $R(G)=0$ on $\omega$. In the present paper, we complete this analysis and provide the derivation of the $\Gamma$-limits $\mathcal{I}_{2n}$ to scaled energies $h^{-2n}\mathcal{E}^{h}$, for all $n\geq 1$. We prove the previously conjectured energy quantisation so that $h^{2n}$ are indeed the only possible scalings, all of them attained (by $G=e^{x_{3}^{n}}Id_{3}$. The structure of $\{\mathcal{I}_{2n}\}_{n\geq 1}$ should be compared with the hierarchy of plate models in the classical nonlinear elasticity [9], as follows. The energy $\mathcal{I}_{2}$ consists of pure bending, quantifying the curvature under the midplate isometric immersion constraint. This is a Kirchhoff-like model, relative to the ambient metric $G$. The next energy $\mathcal{I}_{4}$ consists of linearised first order bending and second order stretching; this is a von Karman-like model, augmented by terms carrying the relevant components of the Riemann tensor $R(G)$. Each higher order energy $\mathcal{I}_{2n}$ consists of linearised bending augmented by the the order-related covariant derivatives of $R(G)$ on the midplate. This is a linear elasticity-like model, in the present context valid in the quantized scaling regimes $n\geq 3$, whereas in the classical case appearing in the regimes $h^{\beta}$ for all $\beta>4$.

Recently, there has been a sustained interest in studying shape formation driven by internal prestrain, through the experimental, modelling via formal methods, numerics, and analytical arguments [36, 18, 14, 7]. General results have been derived in the abstract setting of Riemannian manifolds [20, 19, 32]. Higher energies $\inf\mathcal{E}^{h}\sim h^{\beta}$ with $\beta\in(0,2)$, than the ones analyzed in the present paper may result from the interaction of the metric with boundary conditions or external forces, leading to the “wrinkling-like” effects. Indeed, our setting pertains to the “no wrinkling” regime where $\beta\geq 2$ and the reduced prestrain metric $G_{2\times 2}$ on $\omega$, admits a $W^{2,2}$ isometric immersion in $\mathbb{R}^{3}$. While the systematic description of the singular limits at scalings $\beta<2$ is not yet available, there exists a variety of studies of emerging patterns: compression- driven blistering [15, 3, 4], buckling [10, 11, 12], origami patterns [5, 39], conical singularities [33, 34, 35], or coarsening patterns [1, 2, 38]. In [24, 25, 27], derivations similar to the results of the present paper were carried out under a different assumption on the asymptotic behavior of the prestrain (constant in the present paper), which in particular allowed for the effective energy scalings $h^{\beta}$ in non-even regimes of $\beta>2$. On the frontier of experimental modelling of shape formation, we refer to [17, 16, 40, 21, 13].

1.1. The set-up of the problem

Let $\omega\subset\mathbb{R}^{2}$ be an open, bounded, connected set with Lipschitz boundary. We consider a family of thin hyperelastic sheets occupying the reference domains:

[TABLE]

A typical point in $\Omega^{h}$ is denoted by $x=(x_{1},x_{2},x_{3})=(x^{\prime},x_{3})$. We often use the unit-thickness plate $\Omega^{1}$ as the referential rescaling of each $\Omega^{h}$ via: $\Omega^{h}\ni(x^{\prime},x_{3})\mapsto(x^{\prime},x_{3}/h)\in\Omega^{1}$.

The films $\Omega^{h}$ are characterized by the given smooth incompatibility (Riemann metric) tensor:

[TABLE]

and we want to study the singular limit behaviour, as $h\to 0$, of the following energy functionals:

[TABLE]

defined on vector fields $u^{h}\in{W}^{1,2}(\Omega^{h},\mathbb{R}^{3})$ interpreted as deformations of $\Omega^{h}$. Above, $G(x)^{-1/2}$ stands for the inverse of $G(x)$. When $G=Id_{3}$, the functionals $\mathcal{E}^{h}$ are the classical Hookean nonlinear elastic energies of deformations, with the density $W$ obeying the properties listed below.

In the present general setting, $\mathcal{E}^{h}(u^{h})$ is designed to measure the deviation of $u^{h}$ from being an (equidimensional) isometric immersion of $G$ on $\Omega^{h}$. Indeed, by polar decomposition theorem, $FG^{-1/2}\in SO(3)$ if and only if $F^{\scriptscriptstyle{\mathsf{T}}}F=G$ and $\det F>0$. The Borel-regular, homogeneous density $W:\mathbb{R}^{3\times 3}\rightarrow[0,\infty]$ is thus assumed to satisfy:

• (i)

$W(RF)=W(F)$ for all $R\in{SO}(3)$ and $F\in\mathbb{R}^{3\times 3}$,

• (ii)

$W(F)=0$ for all $F\in{SO}(3)$,

• (iii)

W(F)\geq C\,{\rm dist}^{2}\big{(}F,{SO}(3)\big{)} for all $F\in\mathbb{R}^{3\times 3}$, with some uniform constant $C>0$,

• (iv)

there exists a neighbourhood $\mathcal{U}$ of ${SO}(3)$ such that $W$ is finite and $\mathcal{C}^{2}$ regular on $\mathcal{U}$.

By a more refined analysis [28] one can prove the global counterpart of the above pointwise statement, namely that: $\inf_{W^{1,2}}\mathcal{E}^{h}=0$ if an only if all the components of the Riemann curvature tensor of $G$ vanish identically: $\{R_{ab,cd}\}_{a,b,c,d=1\ldots 3}=0$ on $\Omega^{h}$.

In this paper, we determine the possible energy scalings: $\inf\mathcal{E}^{h}\sim h^{\beta}$ in the limit of vanishing thickness $h\to 0$, and the corresponding variational limits ($\Gamma$-limits) $\mathcal{I}_{\beta}$ of $h^{-\beta}\mathcal{E}^{h}$, in the regime $\beta>4$ that has not been analyzed before. We thus complete the discussion of weakly prestrained films, started in our previous works [28, 6, 29, 30] that covered the range $\beta\in[2,4]$. The singular limits $\mathcal{I}_{\beta}$ are typically given by energies of the form $\mathcal{I}=\|Tensor(y)\|^{2}_{\mathbb{E}}$ defined on the appropriate set of limiting deformations/displacements $y$ of the midplate $\omega$. They quantify the resulting effective curvatures in $Tensor(y)$ relative to $G$ at the level induced by $\beta$, and in the weighted $L^{2}$ norm on $\omega$:

[TABLE]

Above, the quadratic form $\mathcal{Q}_{2}$ carries the two-dimensional reduction of the first nonzero term in the Taylor expansion of $W$ close to its energy well $SO(3)$. More precisely, we define:

[TABLE]

The form $\mathcal{Q}_{3}$ is defined for all $F\in\mathbb{R}^{3\times 3}$, while each $\mathcal{Q}_{2}(x^{\prime},\cdot)$ is defined on $F_{2\times 2}\in\mathbb{R}^{2\times 2}$. Both $\mathcal{Q}_{3}$ and all $\mathcal{Q}_{2}$ are nonnegative definite and depend only on the symmetric parts of their arguments, in view of the assumptions on $W$. The quadratic minimization problem in (1.3) has thus a unique solution among symmetric matrices $\tilde{F}$, which for each $x^{\prime}\in\omega$ is given via the linear function:

[TABLE]

1.2. Description of the main results of this paper

As already pointed out, we will be concerned with the regimes of curvatures of $G$, yielding the incompatibility rate, quantified by $\inf\mathcal{E}^{h}$, of order higher than $h^{4}$ in the thickness $h$. We first recall the following result from [30]:

[TABLE]

The above conditions are further equivalent to existence of smooth vector fields $y_{0},\vec{b}_{1},\vec{b}_{2}:\bar{\omega}\to\mathbb{R}^{3}$, defined uniquely up to rigid motions, such that for the following smooth $\mathbb{R}^{3\times 3}$ matrix fields on $\bar{\omega}$:

[TABLE]

there holds:

[TABLE]

Note that the last equality above implies that we can uniquely define a new smooth vector and matrix fields: $\vec{b}_{3}:\bar{\omega}\to\mathbb{R}^{3}$ and B_{2}=\big{[}\partial_{1}\vec{b}_{2},\leavevmode\nobreak\ \partial_{2}\vec{b}_{2},\leavevmode\nobreak\ \vec{b}_{3}\big{]}, so that: $(B_{0}^{\scriptscriptstyle{\mathsf{T}}}B_{2})_{{\rm sym}\,}+B_{1}^{\scriptscriptstyle{\mathsf{T}}}B_{1}=\frac{1}{2}\partial_{33}G(x^{\prime},0)$. This condition, together with the first two equalities in (1.6) is jointly equivalent to:

[TABLE]

In conclusion, the following three conditions: the two conditions in (1.5) and the one in (1.7), are equivalent. Our first main result generalizes this statement to all even order powers $2(n+1)$ in the infimum energy scaling, for any $n\geq 2$. Moreover, these scalings exhaust all possibilities in the remaining regime: $\inf\mathcal{E}^{h}\sim h^{\beta}$ with $\beta>4$:

Theorem 1.1**.**

The following three statements are equivalent, for each fixed integer $n\geq 2$:

• (i)

$R_{12,12}(x^{\prime},0)=R_{12,13}(x^{\prime},0)=R_{12,23}(x^{\prime},0)=0$* for all $x^{\prime}\in\omega$, and $\partial_{3}^{(k)}R_{i3,j3}(x^{\prime},0)=0$ for all $x^{\prime}\in\omega$, all $k=0\ldots n-2$ and all $i,j=1\ldots 2$.*

• (ii)

$\inf\mathcal{E}^{h}\leq Ch^{2(n+1)}$.

• (iii)

There exist smooth fields $y_{0},\{\vec{b}_{k}\}_{k=1}^{n+1}:\bar{\omega}\to\mathbb{R}^{3}$ such that calling \big{\{}B_{k}=\big{[}\partial_{1}\vec{b}_{k},\leavevmode\nobreak\ \partial_{2}\vec{b}_{k},\leavevmode\nobreak\ \vec{b}_{k+1}\big{]}\big{\}}_{k=1}^{n}, in addition to B_{0}=\big{[}\partial_{1}y_{0},\leavevmode\nobreak\ \partial_{2}y_{0},\leavevmode\nobreak\ \vec{b}_{1}\big{]} satisfying $\det B_{0}>0$, we have:

[TABLE]

or in other words: $\displaystyle\sum_{k=0}^{m}\binom{m}{k}B_{k}^{\scriptscriptstyle{\mathsf{T}}}B_{m-k}-\partial_{3}^{(m)}G(x^{\prime},0)=0\;$ for all $m=0\ldots n$, for all $x^{\prime}\in\omega$.

We further prove compactness and the lower bound, at any of the new viable scaling levels $\inf\mathcal{E}^{h}\sim h^{2(n+1)}$, completing thus the analysis done for $n=0$ in [28, 6] and for $n=1$ in [29, 30]:

Theorem 1.2**.**

Fix $n\geq 2$ and assume that any of the equivalent conditions in Theorem 1.1 holds. Let the sequence of deformations $\{u^{h}\in W^{1,2}(\Omega^{h},\mathbb{R}^{3})\}_{h\to 0}$ satisfy: $\mathcal{E}^{h}(u^{h})\leq Ch^{2(n+1)}$. Then:

• (i)

There exists $\bar{R}^{h}\in SO(3)$, $c^{h}\in\mathbb{R}^{3}$ such that the displacements $\{V^{h}\in W^{1,2}(\omega,\mathbb{R}^{3})\}_{h\to 0}$ in:

[TABLE]

converge as $h\to 0$, strongly in $W^{1,2}(\omega,\mathbb{R}^{3})$, to the limiting displacement:

[TABLE]

• (ii)

The above condition $V\in\mathcal{V}_{y_{0}}$ automatically defines $\vec{p}\in W^{1,2}(\omega,\mathbb{R}^{3})$ such that:

[TABLE]

and then we have: $\displaystyle\liminf_{h\to 0}\frac{1}{h^{2(n+1)}}\mathcal{E}^{h}(u^{h})\geq\mathcal{I}_{2(n+1)}(V)$, where:

[TABLE]

Above, $\mathcal{S}_{y_{0}}$ is the following closed subspace of the Hilbert space $\mathbb{E}$ in (1.2):

[TABLE]

whereas $\mathbb{P}_{\mathcal{S}_{y_{0}}}(F)$ and $\mathbb{P}_{\mathcal{S}_{y_{0}}^{\perp}}(F)$ denote, respectively, the orthogonal projections of $F$ onto the space $\mathcal{S}_{y_{0}}$ and its orthogonal complement $\mathcal{S}_{y_{0}}^{\perp}$ in $\mathbb{E}$. The coefficients in (1.10) are:

[TABLE]

• (iii)

There holds on $\omega$:

[TABLE]

We point out a few related observations:

• (i)

When $G=Id_{3}$, then each functional in (1.10) reduces to the classical linear elasticity. We have: $y_{0}=id$, $\vec{b}_{1}=e_{3}$ and \mathcal{V}=\big{\{}(\alpha x^{\perp}+\vec{\beta},v);\leavevmode\nobreak\ \alpha\in\mathbb{R},\leavevmode\nobreak\ \vec{\beta}\in\mathbb{R}^{2},\leavevmode\nobreak\ v\in W^{2,2}(\omega)\big{\}}, and for $V\in\mathcal{V}$, there holds: $\vec{p}=(-\nabla v,0)$. Consequently: \displaystyle\mathcal{I}_{2(n+1)}(V)=\frac{1}{24}\int_{\omega}\mathcal{Q}_{2}\big{(}x^{\prime},\nabla^{2}v\big{)}{\;\rm d}x^{\prime}, in function of the out-of-plane scalar displacement $v$.

• (ii)

In the present geometric context, the bending term is: $(\nabla y_{0})^{\scriptscriptstyle{\mathsf{T}}}\nabla\vec{p}+(\nabla V)^{\scriptscriptstyle{\mathsf{T}}}\nabla\vec{b}_{1}$. It is of order $h^{n}x_{3}$ and it interacts with the curvature term \big{[}\partial_{3}^{(n-1)}R_{i3,j3}(\cdot,0)\big{]}_{i,j=1\ldots 2}, which is of order $x_{3}^{n+1}$. The interaction occurs only when the two terms have same parity, which happens at even $n$, so $\alpha_{n}=0$ for $n$ odd. The two remaining terms in (1.10) measure the $L^{2}$ norm of \big{[}\partial_{3}^{(n-1)}R_{i3,j3}(\cdot,0)\big{]}_{i,j=1\ldots 2}, with distinct weights assigned to the $\mathcal{S}_{y_{0}}$ and \big{(}\mathcal{S}_{y_{0}}\big{)}^{\perp} projections, again according to the parity of $n$.

• (iii)

The formula in (1.12) relates the quantities appearing in conditions (i) and (iii) of Theorem 1.1. The curvature \big{[}\partial_{3}^{(n-1)}R_{i3,j3}(\cdot,0)\big{]}_{i,j=1\ldots 2} is thus precisely the coefficient of the discrepancy of the order $h^{n+1}$ in (1.8) at the $2\times 2$ minor, scaled by the $(n+1)!/2$ factor.

• (iv)

The finite strain space $\mathcal{S}_{y_{0}}$ can be identtified in the the following two cases.When $y_{0}=id_{2}$, then $\mathcal{S}_{y_{0}}=\{\mathbb{S}\in L^{2}(\omega,\mathbb{R}^{2\times 2}_{{\rm sym}\,});\leavevmode\nobreak\ \mbox{curl}^{\scriptscriptstyle{\mathsf{T}}}\mbox{curl}\,\mathbb{S}=0\}$. When the Gauss curvature \kappa((\nabla y_{0})^{\scriptscriptstyle{\mathsf{T}}}\nabla y_{0})=\kappa\big{(}G_{2\times 2})>0 in $\bar{\omega}$, then $\mathscr{S}_{y_{0}}=L^{2}(\omega,\mathbb{R}^{2\times 2}_{{\rm sym}\,})$, as shown in [26].

Our next result proves the upper bound that is consistent with Theorem 1.1 and yields the $\Gamma$-convergence of the rescaled energies $h^{-2(n+1)}\mathcal{E}^{h}$ to the dimensionally reduced limits $\mathcal{I}_{2(n+1)}$ in (1.10):

Theorem 1.3**.**

Fix $n\geq 2$ and assume that any of the equivalent conditions in Theorem 1.1 hold. Then for every $V\in\mathcal{V}_{y_{0}}$ as defined in (1.9), there exists a sequence $\{u^{h}\in W^{1,2}(\Omega^{h},\mathbb{R}^{3})\}_{h\to 0}$ so that:

[TABLE]

and that: $\displaystyle\liminf_{h\to 0}\frac{1}{h^{2(n+1)}}\mathcal{E}^{h}(u^{h})=\mathcal{I}_{2(n+1)}(V),$ where the limiting energy functional is as in (1.10).

It is worth noting the following self-evident application of Theorems 1.1, 1.2 and 1.3:

Corollary 1.4**.**

Under either of the equivalent conditions in (1.5), assume that for some $n\geq 2$ there holds: \partial_{3}^{(m)}\big{[}R_{i3j3}(\cdot,0)\big{]}_{i,j=1\ldots 2}=0 on $\omega$, for all $m=0\ldots n-2$, but \partial_{3}^{(n-1)}\big{[}R_{i3j3}(\cdot,0)\big{]}_{i,j=1\ldots 2}\not\equiv 0. Then there exist $c,C>0$ such that:

[TABLE]

Moreover, the scaled energies $\displaystyle\frac{1}{h^{2(n-1)}}\mathcal{E}^{h}$, $\Gamma$-converge to the limiting functional $\mathcal{I}_{2(n+1)}$ in (1.10), effectively defined on the space $\mathcal{V}_{y_{0}}$ of first order infinitesimal isometries in (1.9).

For completeness, we note that the conformal metrics of the form: $G(x^{\prime},x_{3})=e^{2\phi(x_{3})}Id_{3}$ provide a class of examples for the viability of all scalings in (1.14). Indeed, the trace midplate metric $e^{2\phi(0)}Id_{2}$ has a smooth isometric immersion $y_{0}=e^{\phi(0)}id_{2}:\omega\to\mathbb{R}^{2}$, and the only possibly nonzero Riemann curvatures of $G$ are given by: $R_{1212}=-\phi^{\prime}(x_{3})^{2}e^{2\phi(x_{3})}$, $R_{1313}=R_{2323}=-\phi^{\prime\prime}(x_{3})e^{2\phi(x_{3})}$. By Corollary 1.4 we see that $\inf\mathcal{E}^{h}\sim h^{2n}$ if and only if $\phi^{(k)}(0)=0$ for $k=1\ldots n-1$ and $\phi^{(n)}(0)\neq 0$.

1.3. The structure of the paper

In sections 2 and 3 we work under the assumption (iii) of Theorem 1.1. First, in Lemma 2.1, we give an easy proof of the implication $(iii)\Rightarrow(ii)$. The particular energy-consistent deformation field can be further used as the local change of variables allowing for the application of the nonlinear rigidity estimate [8] in the present context. This is done in Lemma 2.2 and Corollary 2.3, providing an approximation of an arbitrary energy-consistent deformation gradient $\nabla u^{h}$, by a non-symmetric square root of the $n$-th order Taylor expansion of the metric $G$, derived from the expansion guaranteed in (iii). Both the approximation error and the $L^{2}$ norm of the gradient of the rotation field are energy-controlled. In Lemma 2.4 we prove the compactness part of Theorem 1.2. In Lemma 2.5 we conclude a preliminary lower bound estimate, involving a version of the functional $\mathcal{I}_{2(n+1)}$, whose curvature terms are still expressed in terms of the expansion fields in (iii), as suggested in the right hand side of (1.12).

In section 3, we develop a geometric line of arguments, serving to prove (in Corollary 3.6) the identity (1.12) under assumption (iii). In Lemmas 3.1 and 3.2, we partially reprove the equivalent conditions valid at the previously analyzed scalings $h^{2}$ and $h^{4}$. These statements are then generalized in Lemma 3.5, where we show the implication (iii) $\Rightarrow$ (i), resulting also in the existence of a one order higher approximate field $\vec{b}_{n+1}$, that is given solely through the Christoffel symbols of $G$ on $\omega$.

In section 4 we finally prove Theorem 1.1, showing equivalence of the stated three conditions, by induction on $n\geq 2$. We also finish the proof of Theorem 1.2 by: improving the lower bound from section 2, identifying its curvature components via (1.12), and separating the bending and the excess terms. In section 5 we prove Theorem 1.3, constructing a energy-consistent recovery sequence.

1.4. Notation

Given a matrix $F\in\mathbb{R}^{n\times n}$, we denote its transpose by $F^{\scriptscriptstyle{\mathsf{T}}}$ and its symmetric part by $F_{\mathrm{sym}}=\frac{1}{2}(F+F^{\scriptscriptstyle{\mathsf{T}}})$. The space of symmetric $n\times n$ matrices is denoted by $\mathbb{R}^{n\times n}_{\mathrm{sym}}$, whereas $\mathbb{R}^{n\times n}_{\mathrm{sym,pos}}$ stands for the space of symmetric, positive definite $n\times n$ matrices. By $SO(n)=\{R\in\mathbb{R}^{n\times n};\leavevmode\nobreak\ R^{\scriptscriptstyle{\mathsf{T}}}=R^{-1}\mbox{ and }\det R=1\}$ we mean the group of special rotations; its tangent space at $Id_{n}$ consists of skew-symmetric matrices: $T_{Id_{n}}SO(n)=so(n)=\{F\in\mathbb{R}^{n\times n};\leavevmode\nobreak\ F_{\mathrm{sym}}=0\}$. We use the matrix norm $|F|=(\mbox{trace}(F^{\scriptscriptstyle{\mathsf{T}}}F))^{1/2}$, which is induced by the inner product $\langle F_{1}:F_{2}\rangle=\mbox{trace}(F_{1}^{\scriptscriptstyle{\mathsf{T}}}F_{2})$. The $2\times 2$ principal minor of $F\in\mathbb{R}^{3\times 3}$ is denoted by $F_{2\times 2}$. Conversely, for a given $F_{2\times 2}\in\mathbb{R}^{2\times 2}$, the $3\times 3$ matrix with principal minor equal $F_{2\times 2}$ and all other entries equal to [math], is denoted by $F_{2\times 2}^{*}$. Unless specified otherwise, all limits are taken as the thickness parameter $h$ vanishes: $h\to 0$. By $C$ we denote any universal positive constant, independent of $h$.

1.5. Acknowledgments

M.L. is grateful to Annie Raoult and Shankar Venkataramani for interest in the project and discussions. Support by the NSF grant DMS-1613153 is acknowledged.

2. A proof of Theorem 1.2: compactness and a preliminary lower bound

In this section, assuming condition (iii) of Theorem 1.1, we derive the compactness and (a version of) the lower bound in Theorem 1.2. We first observe the implication $(iii)\Rightarrow(ii)$ in Theorem 1.1:

Lemma 2.1**.**

Assume that condition (iii) in Theorem 1.1 holds, for some $n\geq 2$. Then we have:

[TABLE]

Proof.

Define $\displaystyle u^{h}(x^{\prime},x_{3})=y_{0}+\sum_{k=1}^{n+1}\frac{x_{3}^{k}}{k!}\vec{b}_{k}$, so that:

[TABLE]

Consequently, $(\nabla u^{h})G^{-1/2}$ is positive definite for all small $h$, and modulo a rotation field it equals the following matrix field on $\Omega^{h}$, where we used the assumption (1.8):

[TABLE]

This implies: \displaystyle\mathcal{E}^{h}(u^{h})=\frac{1}{h}\int_{\Omega^{h}}W\big{(}Id_{3}+\mathcal{O}(h^{n+1})\big{)}{\;\rm d}x\leq Ch^{2(n+1)}, as claimed.

Recalling results (1.5) and (1.7) quoted from [30], we already see that $\displaystyle{\lim_{h\to 0}\frac{1}{h^{4}}\inf\mathcal{E}^{h}}=0$ automatically implies: $\inf\mathcal{E}^{h}\leq Ch^{6}$. Before addressing compactness at $h^{2n}$ with $h\geq 3$, we develop the nonlinear rigidity estimates applicable in the present context.

Lemma 2.2**.**

Assume that condition (iii) in Theorem 1.1 holds, for some $n\geq 2$. Let $V\subset\omega$ be an open, Lipschitz subdomain such that $y_{0}$ is injective on $V$. Denote $V^{h}=V\times(-\frac{h}{2},\frac{h}{2})$. Then for every $u^{h}\in W^{1,2}(V^{h},\mathbb{R}^{3})$ there exists $\bar{R}^{h}\in SO(3)$ such that:

[TABLE]

The constant $C$ is uniform for all $V^{h}\subset\Omega^{1}$ that are bi-Lipschitz equivalent with controlled Lipschitz constants.

Proof.

Define $\displaystyle Y=y_{0}+\sum_{k=1}^{n+1}\frac{x_{3}^{k}}{k!}\vec{b}_{k}$, and observe that for $h$ sufficiently small, $Y$ is a smooth diffeomorphism of $V^{h}$ onto its image $U^{h}\subset\mathbb{R}^{3}$. Consider the change of variables $v^{h}=u^{h}\circ Y^{-1}\in W^{1,2}(U^{h},\mathbb{R}^{3})$ and apply the fundamental geometric rigidity estimate [8], yielding existence of $\bar{R}^{h}\in SO(3)$ with:

[TABLE]

Changing variable in the left hand side gives:

[TABLE]

Changing now variable in the right hand side and using (\nabla Y)G^{-1/2}\in SO(3)\big{(}Id_{3}+\mathcal{O}(h^{n+1})\big{)}, as established in Lemma 2.1, results in:

[TABLE]

Combining the three displayed inequalities above proves the result.

The well-known approximation technique [9] together with the arguments in [29, Corollary 2.3], yield the following estimate, whose proof we leave to the reader:

Corollary 2.3**.**

Assume condition (iii) in Theorem 1.1, for some $n\geq 2$. Then, given a sequence $\{u^{h}\in W^{1,2}(\Omega^{h},\mathbb{R}^{3})\}_{h\to 0}$ such that $\mathcal{E}^{h}(u^{h})\leq Ch^{2(n+1)}$, there exists $\{R^{h}\in W^{1,2}(\omega,SO(3))\}_{h\to 0}$ with:

[TABLE]

We now show the compactness part of Theorem 1.1:

Lemma 2.4**.**

Assume condition (iii) in Theorem 1.1, for some $n\geq 2$. Let the sequence of deformations $\{u^{h}\in W^{1,2}(\Omega^{h},\mathbb{R}^{3})\}_{h\to 0}$ satisfy: $\mathcal{E}^{h}(u^{h})\leq Ch^{2(n+1)}$. Then:

• (i)

The averaged displacements $V^{h}$ converge, up to a subsequence, to the first order isometry $V$ as in Theorem 1.2 (i).

• (ii)

The scaled strains \displaystyle\frac{1}{h}\big{(}(\nabla y_{0})^{\scriptscriptstyle{\mathsf{T}}}\nabla V^{h}\big{)}_{{\rm sym}\,} converge, up to a subsequence, weakly in $L^{2}(\omega,\mathbb{R}^{3\times 3})$ to some $\mathbb{S}\in\mathcal{S}_{y_{0}}$.

Proof.

1. Define the following rotation: \displaystyle\bar{R}^{h}=\mathbb{P}_{SO(3)}\fint_{\Omega^{h}}(\nabla u^{h})\Big{(}\sum_{k=0}^{n}\frac{x_{3}^{k}}{k!}B_{k}\Big{)}^{-1}{\;\rm d}x. In order to observe that the above definition is legitimate, we write:

[TABLE]

and upon integrating ${\;\rm d}x^{\prime}$ on the domain $\omega$ while noting Corollary 2.3, obtain:

[TABLE]

Consequently, there also follows:

[TABLE]

Set now $c^{h}\in\mathbb{R}^{3}$ so that $\int_{\omega}V^{h}{\;\rm d}x^{\prime}=0$. We get:

[TABLE]

where we define the following matrix fields whose convergence (up to a subsequence) results from the second bound in (2.1) and from Corollary 2.3:

[TABLE]

We also note that $S\in so(3)$ a.e. in $\omega$. Since the first term in the right hand side of (2.2) converges to [math] in $L^{2}(\omega)$, in virtue of Corollary 2.3, we conclude the following convergence, up to a subsequence:

[TABLE]

It also follows that the limit $S\nabla y_{0}\in W^{1,2}(\omega,\mathbb{R}^{3\times 2})$. A further application of the Poincare inequality to the mean-zero displacements $V^{h}$, yields their strong convergence (up to a subsequence in $W^{1,2}(\omega,\mathbb{R}^{3})$) to some $V\in W^{2,2}(\omega,\mathbb{R}^{3})$ satisfying $\nabla V=(SB_{0})_{3\times 2}$. By skew-symmetry of $S$, it follows that $(\nabla y_{0})^{\scriptscriptstyle{\mathsf{T}}}\nabla V$ is skew a.e. in $\omega$, proving (i).

2. We observe that the first term in the right hand side of (2.2) has its $L^{2}(\omega)$ norm bounded by $Ch^{2}$, in view of the first estimate in Corollary 2.3. Consequently, in the decomposition of \displaystyle\frac{1}{h}\big{(}(\nabla y_{0})^{\scriptscriptstyle{\mathsf{T}}}\nabla V^{h}\big{)}_{{\rm sym}\,}, parallel to that in (2.2), the corresponding first term has a weakly converging subsequence. The remaining second term equals:

[TABLE]

The $L^{2}(\omega)$ norm of the second term above clearly converges to [math], whereas the first term obeys:

[TABLE]

This ends the proof of the claim.

We are now ready to derive the lower bound on the scaled energies $h^{-2(n+1)}\mathcal{E}^{h}(u^{h})$, in terms of the expansion fields $y_{0},\{\vec{b}_{k}\}_{k=1}^{n+1}$ in condition (iii) of Theorem 1.1:

Lemma 2.5**.**

In the context of Lemma 2.4, there holds:

[TABLE]

with the coefficient $\delta_{n+1}$ given by:

[TABLE]

Proof.

1. By Corollary 2.3, the following matrix fields $\{\mathcal{Z}^{h}\in L^{2}(\Omega^{1},\mathbb{R}^{3\times 3})\}_{h\to 0}$ have a converging subsequence, weakly in $L^{2}(\Omega^{1},\mathbb{R}^{3\times 3})$:

[TABLE]

We write: $\displaystyle(R^{h})^{\scriptscriptstyle{\mathsf{T}}}\nabla u^{h}(x^{\prime},hx_{3})=\sum_{k=0}^{n}\frac{h^{k}x_{3}^{k}}{k!}B_{k}+h^{n+1}(R^{h})^{\scriptscriptstyle{\mathsf{T}}}\mathcal{Z}^{h}$ and observe that:

[TABLE]

where the intermediary field $J^{h}$ has the following expansion, on the set $\{|\mathcal{Z}^{h}|^{2}\leq\frac{1}{h}\}\subset\Omega^{1}$:

[TABLE]

Consequently, we get from (2.7) and Taylor expanding $W$ at $Id_{3}$:

[TABLE]

Since $B_{0}^{\scriptscriptstyle{\mathsf{T}}}(R^{h})^{\scriptscriptstyle{\mathsf{T}}}\mathcal{Z}^{h}$ converges weakly in $L^{2}(\Omega^{1},\mathbb{R}^{3\times 3})$, up to a subsequence, to $B_{0}^{\scriptscriptstyle{\mathsf{T}}}\bar{R}^{\scriptscriptstyle{\mathsf{T}}}\mathcal{Z}$, for some $\bar{R}\in SO(3)$ (which is an accumulation point of $\bar{R}^{h}$ in the proof of Lemma 2.4), the above results in:

[TABLE]

2. We need to identify the relevant $2\times 2$ minor of the limiting term \big{(}B_{0}^{\scriptscriptstyle{\mathsf{T}}}\bar{R}^{\scriptscriptstyle{\mathsf{T}}}\mathcal{Z}\big{)}_{{\rm sym}\,} in (2.8). We apply the finite difference technique [9] and consider the following fields $\{f^{s,h}\in W^{1,2}(\Omega^{1},\mathbb{R}^{3})\}_{s>0,h\to 0}$:

[TABLE]

where $S^{h}$ is defined in (2.3). Recall that, as proved in Lemma 2.4, $\nabla V=(SB_{0})_{3\times 2}$ and that $S$ is a.e. in $so(3)$. It follows that the vector $\vec{p}$ defined in Theorem 1.2 (ii) must coincide with $SB_{0}e_{3}=S\vec{b}_{1}$. Consequently, using the first definition above it now easily follows that:

[TABLE]

Using the second definition, we further compute the in-plane derivatives of $f^{s,h}$ for $j=1\ldots 2$:

[TABLE]

The first term in the right hand side above converges to \displaystyle\frac{1}{s}\bar{R}^{\scriptscriptstyle{\mathsf{T}}}\big{(}\mathcal{Z}(x^{\prime},x_{3}+s)-\mathcal{Z}(x^{\prime},x_{3})\big{)}e_{j}, weakly in $L^{2}(\Omega^{1},\mathbb{R}^{3})$, whereas the last two terms may be rewritten as:

[TABLE]

In conclusion, and recalling (2.9), we obtain the following convergence, weakly in $W^{1,2}(\omega,\mathbb{R}^{3})$:

[TABLE]

We thus see that:

[TABLE]

which finally yields:

[TABLE]

3. We now compute the symmetric part of the trace term \big{(}B_{0}^{\scriptscriptstyle{\mathsf{T}}}\bar{R}^{\scriptscriptstyle{\mathsf{T}}}\mathcal{Z}(x^{\prime},0)\big{)}_{2\times 2,{\rm sym}\,} and conclude the proof of the Lemma. It follows from (2.2) and the definition of $\mathcal{Z}^{h}$ in (2.6) that:

[TABLE]

In virtue of (2.6), (2.10) and (2.4), we obtain convergence, weakly in $\mathbb{E}$:

[TABLE]

which allows to conclude, by Lemma 2.4 (ii):

[TABLE]

This ends the proof of Lemma, in virtue of (2.8), (2.10), (2.11) and recalling definitions (1.3).

3. Relations between (i) and (iii) of Theorem

1.1 and a proof of Theorem 1.2 (iii)

In this section we show the relation between the defining quantities appearing in conditions (i) and (iii) of Theorem 1.1. Equivalence of (i) and (iii) at $n=2$ has been shown in [30], building on the previous results in [6, 29], while the proof of the general case will be carried out by induction on $n\geq 2$. We start by introducing some notation that allows for a systematic approach.

Define the smooth matrix fields $\{\Gamma_{a}:\bar{\Omega}^{1}\to\mathbb{R}^{3\times 3}\}_{a=1\ldots 3}$ by setting their coefficients $(\Gamma_{a})_{bc}=\Gamma^{b}_{ac}$ to be the usual Christoffel symbols \displaystyle\Gamma^{b}_{ac}=\frac{1}{2}\sum_{m=1}^{3}G^{bm}\big{(}\partial_{b}G_{mc}+\partial_{c}G_{mb}-\partial_{m}G_{bc}\big{)} of the metric $G$. Recall the standard notation for the coefficients of the inverse: $(G^{-1})_{ab}=G^{ab}$. Since the Levi-Civita connection is torsion-free, it follows that $\Gamma_{a}e_{b}=\Gamma_{b}e_{a}$ for all $a,b=1\ldots 3$ and also, the Riemann curvature tensor is expressed by, for all $c,d=1\ldots 3$:

[TABLE]

Given a matrix field $F:\Omega^{1}\to\mathbb{R}^{3\times 3}$, we define: $\nabla_{a}F=\partial_{a}F+\Gamma_{a}F$ for each $a=1\ldots 3$, so that $(\nabla_{a}F)e_{b}$ coincides with the usual covariant derivative of vector fields: $\nabla_{a}(Fe_{b})$. It also follows that:

[TABLE]

We now partially reprove the mentioned statements at $n=1,2$ for completeness of presentation.

Lemma 3.1**.**

Assume that there exist smooth fields $y_{0},\vec{b}_{1}:\bar{\omega}\to\mathbb{R}^{3}$ such that the matrix field: B_{0}=\big{[}\partial_{1}y_{0},\partial_{2}y_{0},\vec{b}_{1}\big{]} has positive determinant and such that:

[TABLE]

Then:

• (i)

$\partial_{i}B_{0}=B_{0}\Gamma_{i}$* for all $i=1\ldots 2$, and in particular: $\partial_{i}\vec{b}_{1}=B_{0}\Gamma_{3}e_{i}$.*

• (ii)

$R^{a}_{b,ij}(x^{\prime},0)=R_{ab,ij}(x^{\prime},0)=0$* for all $x^{\prime}\in\omega$ and all $a,b=1\ldots 3$, $i,j=1\ldots 2$.*

• (iii)

There exists a unique smooth field $\vec{b}_{2}:\bar{\omega}\to\mathbb{R}^{3}$ such that defining the matrix field B_{1}=\big{[}\partial_{1}\vec{b}_{1},\leavevmode\nobreak\ \partial_{2}\vec{b}_{1},\leavevmode\nobreak\ \vec{b}_{2}\big{]}, there holds: \displaystyle\big{(}B_{0}^{\scriptscriptstyle{\mathsf{T}}}B_{1}\big{)}_{{\rm sym}\,}=\frac{1}{2}\partial_{3}G(x^{\prime},0) for all $x^{\prime}\in\omega$. Moreover:

[TABLE]

Proof.

1. One easily calculates, by a repeated use of the assumed identities, that: $\langle\partial_{i}y_{0},\partial_{j}\vec{b}_{1}\rangle=\partial_{j}G_{i3}-\langle\partial_{ij}y_{0},\vec{b}_{1}\rangle=\partial_{j}G_{i3}-\partial_{i}G_{j3}+\langle\partial_{j}y_{0},\partial_{i}\vec{b}_{1}\rangle$ and thus: $\partial_{3}G_{ij}=\langle\partial_{i}y_{0},\partial_{j}\vec{b}_{1}\rangle+\langle\partial_{j}y_{0},\partial_{i}\vec{b}_{1}\rangle=\partial_{j}G_{i3}-\partial_{i}G_{j3}+2\langle\partial_{j}y_{0},\partial_{i}\vec{b}_{1}\rangle$, for all $i,j=1\ldots 2$, where all the identities are taken on $\omega\times\{0\}$. Thus:

[TABLE]

Secondly: $\langle\partial_{j}y_{0},\partial_{ik}y_{0}\rangle=\partial_{i}G_{jk}-\langle\partial_{k}y_{0},\partial_{ij}y_{0}\rangle=\partial_{i}G_{jk}-\partial_{j}G_{ik}+\langle\partial_{i}y_{0},\partial_{jk}y_{0}\rangle=\partial_{i}G_{jk}-\partial_{j}G_{ik}+\partial_{k}G_{ij}-\langle\partial_{j}y_{0},\partial_{ik}y_{0}\rangle$, which results in:

[TABLE]

Thirdly, from (3.1) we obtain:

[TABLE]

Finally: \langle\vec{b}_{1},\partial_{i}\vec{b}_{1}\rangle=\frac{1}{2}\partial_{i}G_{33}=\big{(}G\Gamma_{i}\big{)}_{33}, so that the last two identities yield:

[TABLE]

This proves (i) and further: $\partial_{i}\vec{b}_{1}=B_{0}\Gamma_{i}e_{3}=B_{0}\Gamma_{3}e_{i}$, as claimed.

2. Using (i) we compute:

[TABLE]

which implies (ii). For (iii), uniqueness of $\vec{b}_{2}$ is obvious, while $\vec{b}_{2}=B_{0}\Gamma_{3}e_{3}$ follows from the requested defining identity, in view of (3.1). The covariant derivative formula is a consequence of (i).

Lemma 3.2**.**

Assume that there exist smooth fields $y_{0},\vec{b}_{1},\vec{b}_{2}:\bar{\omega}\to\mathbb{R}^{3}$ such that the matrix field: B_{0}=\big{[}\partial_{1}y_{0},\leavevmode\nobreak\ \partial_{2}y_{0},\leavevmode\nobreak\ \vec{b}_{1}\big{]} has positive determinant and that together with B_{1}=\big{[}\partial_{1}\vec{b}_{1},\leavevmode\nobreak\ \partial_{2}\vec{b}_{1},\leavevmode\nobreak\ \vec{b}_{2}\big{]} it satisfies:

[TABLE]

Then:

• (i)

$R_{ab,cd}(x^{\prime},0)=0$* for all $x^{\prime}\in\omega$ and all $a,b,c,d=1\ldots 3$.*

• (ii)

There exists a unique smooth field $\vec{b}_{3}:\bar{\omega}\to\mathbb{R}^{3}$ such that defining the matrix field B_{2}=\big{[}\partial_{1}\vec{b}_{2},\leavevmode\nobreak\ \partial_{2}\vec{b}_{2},\leavevmode\nobreak\ \vec{b}_{3}\big{]}, there holds: \displaystyle\big{(}B_{0}^{\scriptscriptstyle{\mathsf{T}}}B_{2}\big{)}_{{\rm sym}\,}+B_{1}^{\scriptscriptstyle{\mathsf{T}}}B_{1}=\frac{1}{2}\partial_{33}G(x^{\prime},0) for all $x^{\prime}\in\omega$. Moreover:

[TABLE]

Proof.

Observe first that for all $a,b=1\ldots 3$ we have:

[TABLE]

Consequently, and using Lemma 3.1 (iii), the last assumed condition is equivalent to:

[TABLE]

The above proves (i), in virtue of Lemma 3.1 (ii) that guarantees $R_{ab,ij}(\cdot,0)=0$ for all $a,b=1\ldots 3$, $i,j=1\ldots 2$. To show (ii), we observe that by Lemma 3.1 and by (i):

[TABLE]

and also, \displaystyle(B_{0}^{\scriptscriptstyle{\mathsf{T}}}B_{2})_{{\rm sym}\,}+B_{1}^{\scriptscriptstyle{\mathsf{T}}}B_{1}-\frac{1}{2}\partial_{33}G(x^{\prime},0)=(B_{0}^{\scriptscriptstyle{\mathsf{T}}}B_{2})_{{\rm sym}\,}+\Gamma_{3}^{\scriptscriptstyle{\mathsf{T}}}G\Gamma_{3}-\Big{(}(G\nabla_{3}\Gamma_{3})_{{\rm sym}\,}+\Gamma_{3}^{\scriptscriptstyle{\mathsf{T}}}G\Gamma_{3}\Big{)}, in view of (3.2), so $B_{2}=B_{0}\nabla_{3}\Gamma_{3}$ satisfies the defining relation. Finally, \partial_{i}\vec{b}_{3}=\partial_{i}\big{(}B_{0}\nabla_{3}\Gamma_{3}e_{3}\big{)}=B_{0}\nabla_{i}\nabla_{3}\Gamma_{3}e_{3} results from Lemma 3.1 (i).

We state the following two useful observations:

Lemma 3.3**.**

For all $n\geq 0$ there holds:

[TABLE]

Proof.

The proof follows by induction. For $n=0$, the statement is obviously true. Assume that it is true for some $n-1$, then:

[TABLE]

Collecting all the terms and recalling that $\binom{n}{k}+\binom{n}{k-1}=\binom{n+1}{k}$ implies the result.

Lemma 3.4**.**

Assume that $R_{12,12}(x^{\prime},0)=R_{12,13}(x^{\prime},0)=R_{12,23}(x^{\prime},0)=0$ for all $x^{\prime}\in\omega$ and also that $\partial_{3}^{(k)}R_{i3,j3}(x^{\prime},0)=0$ for all $k=0\ldots n$, all $i,j=1\ldots 2$ and all $x^{\prime}\in\omega$. Then all the mixed partial derivatives of both $R_{ab,cd}$ and $R^{a}_{b,cd}$, of any order up to $n$, are zero on $\omega$, for all $a,b,c,d=1\ldots 3$.

Proof.

The proof proceeds by induction on $n$. For $n=0$ the result is obviously true. Assume that it is true for some $n\geq 0$ and let the result assumption at $n+1$ hold. Then:

[TABLE]

and we need to show that any partial derivatives of order $n+1$, of the Riemann tensor’s components is zero on $\omega$. This is certainly true for partial derivatives containing $\partial_{i}$ for some $i=1\ldots 2$, so it suffices to prove the claim for $\partial_{3}^{(n+1)}$. Below, we consider various combinations of indices $i,j=1\ldots 2$ and $a,b=1\ldots 3$. Firstly:

[TABLE]

where we used the induction assumption in the first and the third equalities and the second Bianchi identity in the second one. Secondly:

[TABLE]

where we used the induction assumption and (3.3) in the last equality. Thirdly:

[TABLE]

by using (3.4) and the result assumption at $n+1$, in the last equality. Finally: $\partial_{3}^{(n+1)}R_{ab,cd}=0$ by (3.4) and the result assumption.

The following is the main result of this section:

Lemma 3.5**.**

Fix $n\geq 2$. Assume that there exist smooth $y_{0},\{\vec{b}_{k}\}_{k=1}^{n}:\bar{\omega}\to\mathbb{R}^{3}$ such that the matrix fields: B_{0}=\big{[}\partial_{1}y_{0},\leavevmode\nobreak\ \partial_{2}y_{0},\leavevmode\nobreak\ \vec{b}_{1}\big{]} with positive determinant and \{B_{k}=\big{[}\partial_{1}\vec{b}_{k},\leavevmode\nobreak\ \partial_{2}\vec{b}_{k},\leavevmode\nobreak\ \vec{b}_{k+1}\big{]}\}_{k=1}^{n-1}, satisfy:

[TABLE]

Then:

• (i)

Condition in Theorem 1.1 (i) holds.

• (ii)

There exists a unique smooth field $\vec{b}_{n+1}:\bar{\omega}\to\mathbb{R}^{3}$ such that defining the matrix field B_{n}=\big{[}\partial_{1}\vec{b}_{n},\leavevmode\nobreak\ \partial_{2}\vec{b}_{n},\leavevmode\nobreak\ \vec{b}_{n+1}\big{]}, there holds: $\displaystyle\sum_{k=0}^{n}\binom{n}{k}B_{k}^{\scriptscriptstyle{\mathsf{T}}}B_{n-k}=\partial_{3}^{(n)}G(x^{\prime},0)$ for all $x^{\prime}\in\omega$. Moreover:

[TABLE]

Proof.

1. The proof proceeds by induction. The statement at $n=2$ has been shown in Lemma 3.2. We now assume it to be true for some $n\geq 2$. By Lemma 3.4, we get:

[TABLE]

Since B_{k}=\big{[}\partial_{1}\vec{b}_{n},\leavevmode\nobreak\ \partial_{2}\vec{b}_{n},\leavevmode\nobreak\ \vec{b}_{n+1}\big{]} with $\vec{b}_{n+1}$ as in (ii), and recalling Lemma 3.3, we obtain for all $x^{\prime}\in\omega$:

[TABLE]

By (3.6) we can consecutively swap the order of all the covariant derivatives on $\omega\times\{0\}$ in:

[TABLE]

so that:

[TABLE]

In conclusion, using (3.6) again, the formula in (3.7) becomes:

[TABLE]

proving (i) in view of the second assumption at $n+1$.

2. For (ii), observe that $B_{n+1}$ is indeed uniquely defined, by choosing $\vec{b}_{n+2}=B_{n+1}e_{3}$ such that:

[TABLE]

since the principal $2\times 2$ minors of both sides in the above formula coincide by assumption. Further, by (3.8) and the already established (i) at $n+1$, we get:

[TABLE]

Hence, there must be $\vec{b}_{n+1}=B_{0}\nabla_{3}^{(n)}\Gamma_{3}$, as claimed. This ends the proof of the Lemma.

We note that the argument in the proof above leading to (3.9), automatically gives:

Corollary 3.6**.**

For any $n\geq 1$, condition (iii) in Theorem 1.1 implies the formula (1.12).

4. The end of proof of Theorem 1.2 and a proof of Theorem

1.1

The following statement concludes the proof of Theorem 1.2, assuming (iii) of Theorem 1.1:

Lemma 4.1**.**

In the context of Lemma 2.4, there holds: $\displaystyle\liminf_{h\to 0}\frac{1}{h^{2(n+1)}}\mathcal{E}^{h}(u^{n})\geq\mathcal{I}_{2(n+1)}(V)$.

Proof.

By Lemma 2.4 and Corollary 3.6, we get:

[TABLE]

Denoting the $x^{\prime}$-dependent tensor terms at different powers of $x_{3}$ in the integrand above by $I,II$ and $III$, and recalling the definition of $\delta_{n+1}$ in (2.5), the right hand side becomes:

[TABLE]

where by a direct calculation one easily checks that the numerical coefficients $\alpha_{n}$ and $\gamma_{n}$ have the form (1.11). Further, since \mathbb{S}-\delta_{n+1}\big{(}(\nabla y_{0})^{\scriptscriptstyle{\mathsf{T}}}\nabla\vec{b}_{n+1}\big{)}_{{\rm sym}\,}\in\mathcal{S}_{y_{0}}, the first term in the right hand side above is bounded from below by:

[TABLE]

Decomposing the third term into:

[TABLE]

the claim follows by checking that: $\displaystyle\frac{\delta^{2}_{n+1}}{2}+\gamma_{n}=\beta_{n}$ in (1.11).

We are now ready to give:

A proof of Theorem 1.1. The proof is carried out by induction on $n\geq 2$. When $n=2$, then (i) is equivalent with (iii) by facts recalled in the preliminary discussion in section 1.2. Condition (iii) implies (ii) by Lemma 2.1, whereas (ii) implies (i) again in view of (1.5).

Assume now the equivalence of the three conditions at some $n\geq 2$. We want to show the equivalence at $n+1$. Condition (i) implies (iii) by Corollary 3.6. Condition (iii) implies (ii) by Lemma 2.1. Finally, assuming (ii) at $n+1$ allows to write:

[TABLE]

for some infimizing sequence $\{u^{h}\in W^{1,2}(\Omega^{h},\mathbb{R}^{3})\}_{h\to 0}$ and a resulting $V$ from Theorem 1.2. This establishes (i) at $n+1$, in view of the inductive assumption.

For completeness, we state the following auxiliary observations:

Lemma 4.2**.**

In the context of Theorem 1.2, we have:

• (i)

The bending term $(\nabla y_{0})^{\scriptscriptstyle{\mathsf{T}}}\nabla\vec{p}+(\nabla V)^{\scriptscriptstyle{\mathsf{T}}}\nabla\vec{b}_{1}$ is symmetric and it equals:

[TABLE]

• (ii)

Under any of the equivalent conditions in Theorem 1.1 at $n+1$, we have:

[TABLE]

and the following coercivity estimate holds:

[TABLE]

with a constant $C>0$ that depends on $G,\omega$ and $W$ but is independent of $V$.

Proof.

The symmetry of the bending term in (i) follows from:

[TABLE]

The coercivity statement in (ii) has been proved in [30, Theorems 8.2, 8.3].

5. A proof of Theorem 1.3

In this section, we prove the upper bound result of Theorem 1.3. In view of the already established Theorem 1.1, it suffices to show:

Lemma 5.1**.**

Fix $n\geq 2$ and assume condition (iii) in Theorem 1.1. Let $V\in\mathcal{V}_{y_{0}}$ be a first order isometry displacement as in (1.9). Then, there exists a sequence $\{u^{h}\in W^{1,2}(\Omega^{h},\mathbb{R}^{3})\}_{h\to 0}$ of deformations satisfying (1.13), and such that: $\displaystyle\liminf_{h\to 0}\frac{1}{h^{2(n+1)}}\mathcal{E}^{h}(u^{h})=\mathcal{I}_{2(n+1)}(V)$.

Proof.

1. Denote $\displaystyle Y(x^{\prime},x_{3})=y_{0}+\sum_{k=1}^{n+1}\frac{x_{3}^{k}}{k!}\vec{b}_{k}$ and define:

[TABLE]

We now introduce terms in the above expansion. For a fixed small $\varepsilon>0$, the truncated sequence $\{v^{h}\in W^{2,\infty}(\omega,\mathbb{R}^{3})\}_{h\to 0}$ is chosen according to the standard construction in [8] (see also references therein), in a way that:

[TABLE]

The sequence $\{\vec{p}^{h}\in W^{1,\infty}(\omega,\mathbb{R}^{3})\}_{h\to 0}$ is defined by:

[TABLE]

The sequence $\{w^{h}\in\mathcal{C}^{\infty}(\bar{\omega},\mathbb{R}^{3})\}_{h\to 0}$ is such that, recalling (2.5):

[TABLE]

Finally, $\vec{k}_{0}\in\mathcal{C}^{\infty}(\bar{\omega},\mathbb{R}^{3})$ and $\{\vec{q}^{h}\in\mathcal{C}^{\infty}(\bar{\omega},\mathbb{R}^{3})\}_{h\to 0}$, $\{\tilde{r}^{h}\in{L}^{\infty}(\omega,\mathbb{R}^{3})\}_{h\to 0}$ are defined by:

[TABLE]

Further, we choose $\{\vec{r}^{h}\in\mathcal{C}^{\infty}(\bar{\omega},\mathbb{R}^{3})\}_{h\to 0}$ to satisfy, in view of (5.2):

[TABLE]

2. By (5.4) and (5.6) we easily deduce (1.13). Compute now, for all rescaled variables $(x^{\prime},x_{3})\in\Omega^{1}$:

[TABLE]

Consequently, it follows that for $h$ small enough we have:

[TABLE]

which justifies writing, by Taylor’s expansion of $W$ and taking $\epsilon\ll 1$:

[TABLE]

This implies that:

[TABLE]

We thus compute, for all $(x^{\prime},x_{3})\in\Omega^{1}$:

[TABLE]

where:

[TABLE]

3. We now estimate the two last (error) terms in the right hand side of (5.7). Observe that:

[TABLE]

where we have repeatedly used (5.3), (5.4), (5.5) and (5.6), Consequently:

[TABLE]

The first two terms in the right hand side above converge to [math] in $L^{1}(\omega^{1})$ by (5.2) and (5.4). The $L^{1}$ norm of the third term is bounded by $Ch^{4n-2}\|\nabla v^{h}\|_{W^{1,2}}^{4}$ and thus converges to [math] as well. The final fourth term is bounded, in virtue of (5.2) by:

[TABLE]

This completes the convergence analysis of the first error term in (5.7). For the second term, we get:

[TABLE]

As before, the first three terms converge to [math] in $L^{1}(\omega)$, whereas convergence of the last term follows by (5.8). Concluding, and since $\frac{1}{h^{n+1}}\mathcal{R}^{h}$ converges to [math] in $L^{2}(\Omega^{1})$, the limit in (5.7) becomes:

[TABLE]

where for a.e. $(x^{\prime},x_{3})\in\Omega^{1}$ we define:

[TABLE]

In view of (5.8) and since $\|\vec{r}^{h}-\tilde{r}^{h}\|_{L^{2}}$converges to [math] as requested in (5.6), the compatibility in the definition (5.5) now yields from (5.9):

[TABLE]

Now, decomposing the integrand above as in the proof of Lemma 4.1 and recalling convergences in (5.2) and (5.4), we conclude that the right hand side of (5.10) equals $\mathcal{I}_{2(n+1)}(V)$, as claimed.

It is worth observing that directly from Theorems 1.2 and 1.3 we obtain:

Corollary 5.2**.**

Each functional $\mathcal{I}_{2(n+1)}$ attains its infimum and there holds:

[TABLE]

The infima in the left hand side are taken over $W^{1,2}(\Omega,\mathbb{R}^{3})$ deformations $u^{h}$, whereas the minimum in the right hand side is taken over admissible displacements $V\in\mathcal{V}_{y_{0}}$.