Marta Lewicka

AND Annie Raoult
email: annie.raoult@parisdescartes.fr

Diego Ricciotti

Plates with incompatible

Introduction

The purpose of this paper is to study effective elastic behaviour of the incompatibly prestressed thin plates Ω h , characterized by a Riemann metric G given on their reference configuration. The incompatibility is measured through the energy E h given below (sometimes called the "non-Euclidean" elastic energy).

We will be concerned with the regime of curvatures of G which yields the incompatibility rate of order higher than h 2 , in plate's thickness h. Indeed, in paper [START_REF] Bhattacharya | Plates with incompatible prestrain[END_REF] we analyzed the scaling inf E h ∼ h 2 and proved that it only occurs when the metric G 2×2 on the mid-plate can be isometrically immersed in R 3 with the regularity W 2,2 and when, at the same time, the three appropriate Riemann curvatures of G do not vanish identically (for details, see below). The relevant residual theory, obtained through Γ-convergence, yielded a bending Kirchhoff-like residual energy.

In the present paper we assume that:

(1.1)

h -2 inf E h → 0
and prove that the only nontrivial residual theory in this regime is a von Kármán-like energy, valid when inf E h ∼ h 4 . It further turns out that this scaling is automatically implied by (1.1) and inf E h = 0. Indeed, we show that if (1.1) then h -4 inf E h ≤ C, and that h -4 inf E h → 0 if and only if G is immersible whereas trivially min E h = 0 for all h. This scale separation is contrary to the findings of [START_REF] Friesecke | A hierarchy of plate models derived from nonlinear elasticity by Gamma-convergence[END_REF] valid in the Euclidean case of G = Id 3 , where the residual energies are driven by presence of applied forces f h ∼ h α . In that context, three distinct limiting theories have been obtained for E h ∼ h β with β > 2 (equivalently α > 2). Namely: β ∈ (2, 4) corresponded to the linearized Kirchhoff (nonlinear bending) model subject to a nonlinear constraint on the displacements, β = 4 to the classical von Kármán model, and β > 4 to the linear elasticity. The present results are also contrary to the higher order hierarchy of scalings and of the resulting elastic theories of shells, as derived by an asymptotic calculus in [START_REF] Lewicka | The infinite hierarchy of elastic shell models; some recent results and a conjecture, Infinite Dimensional Dynamical Systems[END_REF]. The difference is due to the fact that while the magnitude of external forces is adjustable at will, it seems not to be the case for the interior mechanism of a given metric G which does not depend on h. In fact, it is the curvature tensor of G that induces the nontrivial stresses in the thin film. The Riemann tensor of a three-dimensional metric has only six independent components, namely the six sectional curvatures created out of the three principal directions, which further fall into two categories: including or excluding the thin direction variable. The simultaneous vanishing of curvatures in each of such categories correspond to the two scenarios at hand in terms of the scaling of the residual energy.

1.1. Some background in dimension reduction for thin structures. Early attempts for replacing the three-dimensional model of a thin elastic structure with planar mid-surface at rest, by a two-dimensional model, were based on a priori simplifying assumptions on the deformations and on the stresses. Later, the natural idea of using the thickness as a small parameter and of establishing a limit model was largely explored; we refer in particular to the works by Ciarlet and Destuynder who set the method in the appropriate framework of the weak formulation of the boundary value problems [START_REF] Ciarlet | A justification of the two-dimensional linear plate model[END_REF][START_REF] Ciarlet | A justification of a nonlinear model in plate theory[END_REF], proved convergence to the linear plate model [START_REF] Destuynder | Comparaison entre les modèles tri-dimensionnels et bi-dimensionnels de plaques en élasticité[END_REF] in the context of linearized elasticity, and obtained formally the von Kármán plate model from finite elasticity [START_REF] Ciarlet | A justification of the von Kármán equations[END_REF]. See also [START_REF] Raoult | Construction d'un modèle d'évolution de plaques avec terme d'inertie de rotation[END_REF][START_REF] Raoult | Analyse mathématique de quelques modèles de plaques et de poutres élastiques ou élastoplastiques[END_REF] for the time-dependent case and convergence results, and [START_REF] Ciarlet | Mathematical Elasticity[END_REF] for a comprehensive list of references.

The issue of deriving two-dimensional models valid for large deformations, by means of an asymptotic formalism, was subsequently tackled by Fox, Simo and the second author in [START_REF] Fox | A justification of nonlinear properly invariant plate theories[END_REF]. They showed, in the context of the Saint Venant-Kirchhoff materials subject to appropriate boundary conditions, how to recover a hierarchy of four models. This hierarchy, driven by the order of magnitude of the applied loads, consisted of: the nonlinear membrane model, the inextensional bending model, the von Kármán model and the linear plate model. The models thus obtained still required a justification through rigorous convergence results. In [START_REF] Dret | The nonlinear membrane model as a variational limit of nonlinear threedimensional elasticity[END_REF], Le Dret and the second author used the variational point of view and proved Γ-convergence of the three-dimensional elastic energies to a nonlinear membrane energy, valid for loads of magnitude of order 1. We remark that the expression of the limiting stored energy therein consisted of quasiconvexification of the three-dimensional energy, first minimized with respect to the normal stretches. This allowed to recover the degeneracy under compression; a feature that is otherwise missed by formal expansions. We further mention that for 3d→ 1d reduction a similar point of view had been introduced by Acerbi, Buttazzo and Percivale in [START_REF] Acerbi | A variational definition for the strain energy of an elastic string[END_REF].

A key-point for deriving rigorously the above mentioned nonlinear bending model has been the geometric rigidity result due to Friesecke, James and Müller [START_REF] Friesecke | A theorem on geometric rigidity and the derivation of nonlinear plate theory from three dimensional elasticity[END_REF]. In the similar spirit, the same authors justified the von Kármán model, the linear model [START_REF] Friesecke | A hierarchy of plate models derived from nonlinear elasticity by Gamma-convergence[END_REF] and also they introduced novel intermediate models, in particular in the range of energies -or equivalently of loadings -between the scaling responsible for bending (β = 2) and the von Kármán scaling (β = 4). In this range of models, the three-dimensional stored energy appears in the limit stored energy through its second derivative at rest. Scaling the energy with exponents β other than integers had been explored for the membrane to bending range in [START_REF] Conti | Confining thin sheets and folding paper[END_REF] leading to convergence results for 0 < β < 5/3 while the regime 5/3 ≤ β < 2 remains open and is conjectured to be relevant for crumpling of elastic sheets. Other significant extensions concern derivation of limit theories for incompressible materials [START_REF] Conti | Derivation of elastic theories for thin sheets and the constraint of incompressibility[END_REF][START_REF] Conti | Derivation of a plate theory for incompressible materials[END_REF][START_REF] Trabelsi | Modeling of a membrane for nonlinearly elastic incompressible materials via Gammaconvergence[END_REF][START_REF] Li | The von Karman theory for incompressible elastic shells[END_REF], for heterogeneous materials [START_REF] Schmidt | Plate theory for stressed heterogeneous multilayers of finite bending energy[END_REF], through establishing convergence of equilibria rather than strict minimizers [START_REF] Monneau | Justification of nonlinear Kirchhoff-Love theory of plates as the application of a new singular inverse method[END_REF][START_REF] Sharon | Geometrically driven wrinkling observed in free plastic sheets and leaves[END_REF][START_REF] Lewicka | A note on the convergence of low energy critical points of nonlinear elasticity functionals, for thin shells of arbitrary geometry[END_REF][START_REF] Mora | Convergence of equilibria of thin elastic plates under physical growth conditions for the energy density[END_REF][START_REF] Lewicka | Convergence of equilibria for incompressible elastic plates in the von Karman regime[END_REF], and finally for shallow shells [START_REF] Lewicka | Models for elastic shells with incompatible strains[END_REF].

Extension of the above variational method valid in the framework of the large deformation model was conducted in parallel for slender structures whose midsurface at rest is non-planar. The first result by the second author and Le Dret [START_REF] Dret | The membrane shell model in nonlinear elasticity: a variational asymptotic derivation[END_REF] relates to scaling β = 0 and models membrane shells: the limit stored energy depends then only on the stretching and shearing produced by the deformation on the midsurface. Another study is due to Friesecke, James, Mora and Müller [START_REF] Friesecke | Derivation of nonlinear bending theory for shells from three-dimensional nonlinear elasticity by Gamma-convergence[END_REF] who analyzed the case β = 2. This scaling corresponds to a flexural shell model, where the only admissible deformations are those preserving the midsurface metric. The limit energy depends then on the change of curvature produced by the deformation. Further, the first author, Mora and Pakzad derived the relevant linear theories (β > 4) and the von Kármán-like theories (β = 4) in [START_REF] Lewicka | Shell theories arising as low energy Gamma-limit of 3d nonlinear elasticity[END_REF], and subsequently proceeded to finalize the analysis for elliptic shells in the full regime β > 2 in [START_REF] Lewicka | The matching property of infinitesimal isometries on elliptic surfaces and elasticity of thin shells[END_REF]. A similar analysis has been performed in case of the developable shells in [START_REF] Hornung | Infinitesimal isometries on developable surfaces and asymptotic theories for thin developable shells[END_REF] leading to the proof of the collapse of all residual theories to the linear theory when β > 2. Following these findings, a conjecture was made in [START_REF] Lewicka | The infinite hierarchy of elastic shell models; some recent results and a conjecture, Infinite Dimensional Dynamical Systems[END_REF] about the infinite hierarchy of shell models and the various possible limiting scenarios differentiatied by rigidity properties of shells. Let us recall that a comprehensive body of work had been previously devoted to the asymptotic derivation of shell models in the small displacement regime under clear hypotheses on the model taken for granted, threedimensional or already two-dimensional and containing the thickness as a parameter. Several models were recovered by Ciarlet and coauthors [START_REF] Ciarlet | Asymptotic analysis of linearly elastic shells. I. Justification of membrane shells equations[END_REF][START_REF] Ciarlet | Asymptotic analysis of linearly elastic shells : Generalized membrane shells[END_REF][START_REF] Ciarlet | Asymptotic analysis of linearly elastic shells. II. Justification of flexural shells[END_REF], by Destuynder [START_REF] Destuynder | Sur une justification des modèles de plaques et de coques par les méthodes asymptotiques[END_REF][START_REF] Destuynder | A classification of thin shell theories[END_REF] and by Sanchez-Palencia and coauthors [START_REF] Sanchez-Palencia | Statique et dynamique des coques minces. I. Cas de flexion pure non inhibée[END_REF][START_REF] Sanchez-Palencia | Statique et dynamique des coques minces. II. Cas de flexion pure inhibée. Approximation membranaire[END_REF][START_REF] Caillerie | Elastic thin shells: asymptotic theory in the anisotropic and heterogeneous cases[END_REF][START_REF] Caillerie | A new kind of singular stiff problems and application to thin elastic shells[END_REF][START_REF] Miara | Asymptotic analysis of linearly elastic shells[END_REF]. Sanchez-Palencia, in particular, theorized the role and interplay of the midsurface geometry and of the boundary conditions [START_REF] Sanchez-Palencia | Passage à la limite de l'élasticité tridimensionnelle à la théorie asymptotique des coques minces[END_REF], as well as underlined the singular perturbation behavior. We refer to [START_REF] Ciarlet | Mathematical Elasticity[END_REF] for many additional references.

Most recently, there has been a sustained interest in studying similar problems where the shape formation is not driven by exterior forces but rather by the internal prestrain caused by e.g. growth, swelling, shrinkage or plasticity [START_REF] Klein | Shaping of elastic sheets by prescription of non-Euclidean metrics[END_REF][START_REF] Efrati | Elastic theory of unconstrained non-Euclidean plates[END_REF][START_REF] Sharon | Geometrically driven wrinkling observed in free plastic sheets and leaves[END_REF]. Variants of a thin plate theory can be used to study the self-similar structures which form due to variations in an intrinsic metric of a surface that is asymptotically flat at infinity [START_REF] Audoly | Self-similar structures near boundaries in strained systems[END_REF], and also in the case of a circular disk with edge-localized growth [START_REF] Efrati | Elastic theory of unconstrained non-Euclidean plates[END_REF], or in the shape of a long leaf [START_REF] Liang | The shape of a long leaf[END_REF]. Ben Amar and coauthors formally derived a variant of the Föppl-von Kármán equilibrium equations from finite incompressible elasticity [START_REF] Dervaux | Morphogenesis of growing soft tissues[END_REF][START_REF] Dervaux | Morphogenesis of thin hyperelastic plates: a constitutive theory of biological growth in the Foppl-von Karman limit[END_REF]: they use the multiplicative decomposition of the gradient proposed in [START_REF] Rodriguez | S tress-dependent finite growth in finite soft elatic tissues[END_REF] similar to ours. They take cockling of paper, grass blades and sympatelous flowers as examples [START_REF] Dervaux | Morphogenesis of thin hyperelastic plates: a constitutive theory of biological growth in the Foppl-von Karman limit[END_REF][START_REF] Ben Amar | Petal shapes of sympetalous flowers: the interplay between growth, geometry and elasticity[END_REF].

A systematic study of the possible limit problems when a target metric is prescribed was undertaken by the first author and collaborators: a residually strained version of the Kirchhoff theory for plates was, for the first time, rigorously derived in [START_REF] Lewicka | Scaling laws for non-Euclidean plates and the W 2,2 isometric immersions of Riemannian metrics[END_REF] under the assumption that the target metric is independent of thickness. This analysis was completed in [START_REF] Bhattacharya | Plates with incompatible prestrain[END_REF], resulting in a necessary and sufficient condition that the elastic prestrained energy scales as h 2 . The object of the present paper is to study higher order prestrains.

Let us also mention that in [START_REF] Lewicka | Models for elastic shells with incompatible strains[END_REF][START_REF] Lewicka | The Monge-Ampere constrained elastic theories of shallow shells[END_REF][START_REF] Lewicka | Variational models for prestrained plates with Monge-Ampere constraint[END_REF] similar derivations were carried out under a different assumption on the asymptotic behavior of the prescribed metric, which also implied energy scaling h β in different regimes of β > 2. In [START_REF] Lewicka | Models for elastic shells with incompatible strains[END_REF] it was shown that the resulting equations are identical to those postulated to account for the effects of growth in elastic plates [START_REF] Liang | The shape of a long leaf[END_REF] and used to describe the shape of a long leaf. In [START_REF] Lewicka | Variational models for prestrained plates with Monge-Ampere constraint[END_REF] a model with a Monge-Ampère constraint was derived and analysed from various aspects. Other results concerning the energy scaling for the materials with residual strain are derived in [START_REF] Bella | Metric-induced wrinkling of a thin elastic sheet[END_REF], where by imposing suitable boundary data, conditions of [START_REF] Lewicka | Scaling laws for non-Euclidean plates and the W 2,2 isometric immersions of Riemannian metrics[END_REF][START_REF] Bhattacharya | Plates with incompatible prestrain[END_REF] are not satisfied and hence the residual energy scales larger than h 2 , depending on the type of these boundary data (see also [START_REF] Sharon | Geometrically driven wrinkling observed in free plastic sheets and leaves[END_REF]).

1.2. The set-up and notation. Let Ω be an open, bounded, smooth and simply connected subset of R 2 . For 0 < h 1 we consider thin films Ω h around the mid-plate Ω:

(1.2)

Ω h = x = (x , x 3 ); x ∈ Ω, x 3 ∈ (-h/2, h/2) .
Let G : Ωh → R 3×3 be a given smooth Riemann metric on Ω h , uniform through the thickness:

G(x , x 3 ) = G(x ) for every (x , x 3 ) ∈ Ω h ,
and let A = √ G denote the unique positive definite symmetric square root of G. Consider the energy functional E h : W 1,2 (Ω h , R 3 ) → R+ defined as:

(1.3) E h (u h ) = 1 h ˆΩh W (∇u h A -1 ) dx.
The nonlinear elastic energy density W : R 3×3 → R+ is a Borel measurable function, assumed to be C 2 in a neighborhood of SO(3) and to satisfy, for every

F ∈ R 3×3 , every R ∈ SO(3)
and with a uniform constant c > 0, the conditions:

W (R) = 0, W (RF ) = W (F ), W (F ) ≥ c dist 2 (F, SO(3)) . (1.4)
The first condition states that the energy of a rigid motion is 0, while the second is the frame invariance. They imply that DW (Id 3 ) = 0 and that D 2 W (Id 3 )(S, •) = 0 for all skew symmetric matrices S ∈ so(3). The third assumption above reflects the quadratic growth of the density W away from the energy well SO(3). Note that these assumptions are not contradictory with the physical condition W (F ) = ∞ for det F ≤ 0.

Throughout the paper, we use the following notation. Given a matrix F ∈ R 3×3 , we denote its transpose by F t , its symmetric part by symF = 1 2 (F + F t ), and its skew part by skewF = F -symF . By SO(n) = {R ∈ R n×n ; R t = R -1 and det R = 1} we denote the group of special rotations, while so(n) = {F ∈ R n×n ; symF = 0} is the space of skewsymmetric matrices. We use the matrix norm |F | = (trace(F t F )) 1/2 , which is induced by the inner product F 1 : F 2 = trace(F t 1 F 2 ). The 2 × 2 principal minor of a matrix F ∈ R 3×3 is denoted by F 2×2 . Conversely, for a given F 2×2 ∈ R 2×2 , the 3 × 3 matrix with principal minor equal F 2×2 and all other entries equal to 0, is denoted by (F 2×2 ) * . All limits are taken as the thickness parameter h vanishes, i.e. when h → 0. Finally, by C we denote any universal constant, independent of h.

1.3. Some previous directly related results. It has been proved in [START_REF] Lewicka | Scaling laws for non-Euclidean plates and the W 2,2 isometric immersions of Riemannian metrics[END_REF] that:

inf u h ∈W 1,2 (Ω h ,R 3 ) E h (u h ) = 0
if and only if the Riemann curvature tensor of G vanishes identically in Ω h , i.e.: Riem(G) ≡ 0, and when (equivalently) the infimum above is achieved through a smooth isometric immersion u h of the metric G on Ω h . Further, in [START_REF] Bhattacharya | Plates with incompatible prestrain[END_REF] it is proved that: lim h→0 1 h 2 inf E h = 0 if and only if the following Riemann curvatures of G vanish identically:

(1.5) R 1212 = R 1213 = R 1223 ≡ 0 in Ω h .
More generally, the limit behavior of the rescaled energies h -2 E h has been investigated in [START_REF] Bhattacharya | Plates with incompatible prestrain[END_REF] and it has been proved that their Γ-limit is given by the functional:

I 2 (y) = 1 24 ˆΩ Q 2,A x , (∇y) t ∇ b dx ,
effectively defined on the set of all y ∈ W 2,2 (Ω, R 3 ) such that (∇y) t ∇y = G 2×2 . The quadratic forms Q 2,A (x , •) are given by means of the energy density W as in (3.4). The Cosserat vector b ∈ W 1,2 ∩ L ∞ (Ω, R 3 ) is uniquely determined from the isometric immersion y by:

(1.6)

Q t Q = G where Qe 1 = ∂ 1 y, Qe 2 = ∂ 2 y, Qe 3 = b, with det Q > 0.
Observe that the functional I 2 is a Kirchhoff-like fully nonlinear bending energy, which in case of Ge 3 = e 3 reduces to the classical bending content quantifying the second fundamental form (∇y) t ∇ b = (∇y) t ∇ N on the deformed surface y(Ω) with the unit normal vector N .

We recall that by Theorems 5.3, 5.5 and Corollary 5.4 in [START_REF] Bhattacharya | Plates with incompatible prestrain[END_REF], the negation of condition (1.5) is equivalent to min I 2 > 0. For this reason, (1.5) is equivalent to the existence of a, necessarily unique and smooth, isometric immersion y 0 : Ω → R 3 of G 2×2 , such that:

(1.7) (∇y 0 ) t ∇y 0 = G 2×2 sym (∇y 0 ) t ∇ b 0 = 0.
Above, the smooth vector field b 0 and the smooth matrix field Q 0 are given as in (1.6):

(1.8)

Q t 0 Q 0 = G, Q 0 e 1 = ∂ 1 y 0 , Q 0 e 2 = ∂ 2 y 0 and Q 0 e 3 = b 0 with det Q 0 > 0. Equivalently, denoting the inverse matrix G -1 = [G ij ] i,j:1..3 , we have: (1.9) b 0 = - 1 G 33 G 13 ∂ 1 y 0 + G 23 ∂ 2 y 0 + 1 √ G 33 N .
Uniqueness of the immersion y 0 in (1.7) follows from Theorem 5.3 in [START_REF] Bhattacharya | Plates with incompatible prestrain[END_REF] which shows that the second fundamental form of the surface y 0 (Ω) is given in terms of G. Therefore, both fundamental forms are known. Also, the second equation in (1.7) comes from the fact that the kernel of each quadratic form Q 2,A consists of so(2).

1.4. New results of this work. In view of the above statements, in this paper we investigate smaller energy scalings and the limiting behaviour of the minimizing configurations to E h under condition (1.5). We first prove (in Lemma 2.1) that (1.5), which as we recall is equivalent to (1.1), implies: inf E h ≤ Ch 4 .

We then derive (in Theorem 3.1 and Theorem 4.1) the Γ-limit I 4 of the rescaled energies h -4 E h , together with their compactness properties. Namely, let y 0 be the unique immersion satisfying (1.7), where b 0 is as in (1.8). Let d 0 : Ω → R 3 be the smooth vector field given in terms of y 0 by:

(1.10) Q t 0 d 0 , e 1 = -∂ 1 b 0 , b 0 , Q t 0 d 0 , e 2 = -∂ 2 b 0 , b 0 , Q t 0 d 0 , e 3 = 0.
The limit I 4 is then the following energy functional:

I 4 (V, S) = 1 2 ˆΩ Q 2,A x , S + 1 2 (∇V ) t ∇V + 1 24 (∇ b 0 ) t ∇ b 0 dx + 1 24 ˆΩ Q 2,A x , (∇y 0 ) t ∇ p + (∇V ) t ∇ b 0 dx + 1 1440 ˆΩ Q 2,A x , (∇y 0 ) t ∇ d 0 + (∇ b 0 ) t ∇ b 0 dx , (1.11)
acting on the space of finite strains:

S ∈ cl L 2 sym (∇y 0 ) t ∇w); w ∈ W 1,2 (Ω, R 3
and the space of first order infinitesimal isometries:

V ∈ W 2,2 (Ω, R 3 ) such that: sym (∇y 0 ) t ∇V ) 2×2 = 0,
where the vector field p ∈ W 1,2 (Ω, R 3 ) is uniquely associated with each V by: (∇y 0 ) t p = -(∇V ) t b 0 and b 0 , p = 0. The spaces consisting of S and V contain the information about the admissible error displacements, relative to the leading order immersion y 0 , under the energy scaling E h ∼ h 4 . We discuss the geometrical significance of V and S and of various bending and stretching tensors in the first two terms of I 4 (V, S) in section 5. We further prove in Theorem 6.2 that the last term in (1.11), which is obviously constant and as such does not play a role in the minimization process, is precisely given by the only potentially nonzero (in view of (1.5)) curvatures of G, namely:

sym (∇y 0 ) t ∇ d 0 + (∇ b 0 ) t ∇ b 0 = R 1313 R 1323 R 1323 R 2323 .
We may thus write, informally:

I 4 (V, S) = 1 2 ˆΩ Q 2,A (x , stretching of order h 2 ) dx + 1 24 ˆΩ Q 2,A (x , bending of order h) dx + 1 1440 ˆΩ Q 2,A (x , Riemann curvature of G) dx .
In particular, since all three terms above are nonnegative, this directly implies that the condition lim h→0 1 h 4 inf E h = 0, which is equivalent to min I 4 = 0, is further equivalent to the immersability G, i.e. the vanishing of all its Riemann curvatures Riem(G) ≡ 0 in Ω h . 1.5. Acknowledgments. M.L. was partially supported by the NSF grant DMS-0846996 and the NSF grant DMS-1406730. A part of this work has been carried out while the first author visited the second author at the Université Paris Descartes, whose support and warm hospitality are gratefully acknowledged.

The scaling and approximation lemmas

We first introduce the following notation. Let B 0 (x ) be the matrix field satisfying:

(2.1)

B 0 e 1 = ∂ 1 b 0 , B 0 e 2 = ∂ 2 b 0 and B 0 e 3 = d 0 ,
where d 0 is given by (1.10). Observe that in this way Q t 0 B 0 is skew symmetric. Indeed, it has the following block form:

(2.2) Q t 0 B 0 = (∇y 0 ) t ∇ b 0 (∇y 0 ) t d 0 ( b 0 ) t ∇ b 0 b 0 , d 0 and by (1.7) we see that (∇y 0 ) t ∇ b 0 ∈ so(2) is skew symmetric, while by (1.10): (∇y 0 ) t d 0 = -(∇ b 0 ) t b 0 and b 0 , d 0 = 0. Lemma 2.1. Condition (1.5) implies: inf W 1,2 (Ω h ,R 3 ) E h ≤ Ch 4 .
Proof. Let us construct a sequence u h ∈ W 1,2 (Ω h , R 3 ) that has low energy. Let:

(2.3) u h (x , x 3 ) = y 0 (x ) + x 3 b 0 (x ) + x 2 3 2 d 0 (x ),
in fact each u h is the restriction on its domain Ω h of the same deformation. We have:

∇u h (x , x 3 ) = Q 0 (x ) + x 3 B 0 (x ) + x 2 3 2 D 0 (x ),
where the matrix field D 0 (x ) ∈ R 3×3 is given through:

D 0 (x )e 1 = ∂ 1 d 0 , D 0 (x )e 2 = ∂ 2 d 0 , D 0 (x )e 3 = 0,
so that:

∇u h A -1 = Q 0 A -1 + x 3 B 0 A -1 + x 2 3 2 D 0 A -1 .
For brevity, denote

F h = ∇u h A -1 .
Obviously, F h decomposes as:

(2.4)

F h (x , x 3 ) = Q 0 A -1 (x )(Id 3 + x 3 S(x ) + x 2 3 T (x )) = (Q 0 A -1 (x ))G h (x , x 3 ) with S = A -1 Q t 0 B 0 A -1 , T = 1 2 A -1 Q t 0 D 0 A -1 and G h = Id 3 + x 3 S + x 2 3 T . Since Q 0 A -1 ∈ SO(3) by construction, frame indifference implies that W (F h ) = W ((G h ) t G h ) 1/2 . Note that S is skew symmetric, because Q t 0 B 0 is skew symmetric. Therefore, (G h ) t G h
and the expansion of its square root do not contain terms linear in x 3 . Indeed, letting

K = T +T t -S 2 : ((G h ) t G h )(x , x 3 ) = Id 3 + x 2 3 K(x ) + O(x 3
3 ) and:

((G h ) t G h ) 1/2 (x , x 3 ) = Id 3 + x 2 3 2 K(x ) + O(x 3 3 ).
As a consequence, using W (Id 3 ) = 0 and DW (Id 3 ) = 0, we obtain:

W (F h ) = W ((G h ) t G h ) 1/2 = x 4 3 8 D 2 W (Id 3 )(K, K) + O(x 5
3 ).

Using (1.3), we get

E h (u h ) = 1 h ˆΩh W (F h ) dx ≤ Ch 4 ,
which accomplishes the proof of the lemma.

In Lemma 2.1 above, we constructed deformations whose gradient was sufficiently close to Q 0 + x 3 B 0 , to provide the energy of the order h 4 . Conversely, in Corollary 2.3 below, we establish that the gradients of deformations u h whose energy scales like h 4 , are close to Q 0 + x 3 B 0 modulo local multiplications by R h (x ) ∈ SO(3). Corollary 2.3 makes this statement precise and gives an estimation on ∇R h as well.

For any V which is an open subset of Ω, we let V h = V × (-h/2, h/2) and we define the local energy functional by:

E h (u h , V h ) = 1 h ˆVh W (∇u h A -1 ) dx.
Lemma 2.2. Assume (1.5). There exists a constant C > 0 with the following property. For any u h ∈ W 1,2 (V h , R 3 ), there exists Rh ∈ SO(3) such that:

(2.5) 1 h ˆVh ∇u h (x) -Rh (Q 0 (x ) + x 3 B 0 (x )) 2 dx ≤ C E h (u h , V h ) + h 3 |V h | .
The constant C is uniform for all V h which are bi-Lipschitz equivalent with controlled Lipschitz constants.

Proof. By assumption (1.4), we have:

(2.6) E h (u h , V h ) ≥ c h ˆVh dist 2 ∇u h A -1 , SO(3) dx.
This suggests performing a change of variables in order to use the nonlinear geometric rigidity estimate [START_REF] Friesecke | A hierarchy of plate models derived from nonlinear elasticity by Gamma-convergence[END_REF]. For any

u h ∈ W 1,2 (V h , R 3 ), we let v h = u h • Y -1 with Y : V h → Y (V h ) = U h ⊂ R 3 given as in (2.
3), namely:

Y (x , x 3 ) = y 0 (x ) + x 3 b 0 (x ) + x 2 3 2 d 0 (x ). Obviously, v h ∈ W 1,2 (U h , R 3 ) and:
(2.7)

∇u h A -1 (x , x 3 ) = ∇v h (z)(∇Y A -1 )(x , x 3 ), z := Y (x , x 3 ). Let S = B 0 Q -1 0 and T = 1 2 D 0 Q -1 0 . Note that S = B 0 Q -1 0 = Q -1,t 0 (Q t 0 B 0 Q -1 0 ) = -Q -1,t 0 B t 0 in view of Q t 0 B 0 ∈ so(3)
. Therefore S ∈ so(3). Computations as in Lemma 2.1 now give:

(2.8) ∇Y (x , x 3 ) = Q 0 (x ) + x 3 B 0 (x ) + x 2 3 2 D 0 (x ),
and:

∇Y A -1 = Id 3 + x 3 S (x ) + x 2 3 T (x ) (Q 0 A -1
). We see that for h small, det(∇Y A -1 ) > 0. Further, the left polar decomposition ∇Y A -1 = ∇Y A -1 (∇Y A -1 ) t 1/2 R, allows us to write: 1) is a symmetric matrix field and R ∈ SO(3). Again, the symmetric term does not contain any term linear in x 3 . Therefore:

∇Y A -1 = (Id 3 + x 2 3 M (x , x 3 ))R(x , x 3 ), where M = O(
dist ∇v h ∇Y A -1 , SO(3) = dist ∇v h (Id 3 + x 2 3 M )R, SO(3) = dist ∇v h (Id 3 + x 2 3 M ), SO(3) ≥ c dist ∇v h , SO(3)(Id 3 + x 2 3 M ) -1 ≥ c dist ∇v h , SO(3) + O(x 2 3 ). Now, let J = det ∇Y • Y -1 -1
. By (2.7) and the above computation:

ˆVh dist 2 ∇u h A -1 , SO(3) dx ≥ c ˆUh dist 2 ∇v h , SO(3) J dz -c ˆVh x 4 3 dx.
In other words, since J ≥ c > 0:

1 h ˆVh dist 2 ∇u h A -1 , SO(3) dx + h 3 |V h | ≥ c h ˆUh dist 2 ∇v h , SO(3) dz.
By [START_REF] Friesecke | A hierarchy of plate models derived from nonlinear elasticity by Gamma-convergence[END_REF], there exists C > 0 with the following property. For any v h ∈ W 1,2 (U h , R 3 ), there exists Rh ∈ SO(3) such that:

C ˆUh dist 2 ∇v h , SO(3) dz ≥ ˆUh ∇v h -Rh 2 dz.
The constant C can be chosen uniformly for domains U h which are bi-Lipschitz equivalent with controlled Lipschitz constants. By (2.6) and the reverse change of variables which satisfies J -1 ≥ c > 0 and |∇Y | ≤ C, we obtain:

C E h (u h , V h ) + h 3 |V h | ≥ 1 h ˆVh ∇u h -Rh ∇Y 2 dx,
again with a constant C uniform for domains V h that are bi-Lipschitz equivalent with controlled Lipschitz constants. This accomplishes the proof in view of (2.8).

Corollary 2.3. Assume (1.5) and let u h be a sequence of deformations such that:

lim h→0 h -2 E h (u h ) = 0.
Then, there exist matrix fields R h ∈ W 1,2 (Ω, SO(3)) such that:

(2.9)

1 h ˆΩh ∇u h (x) -R h (x ) Q 0 (x ) + x 3 B 0 (x ) 2 dx ≤ C E h (u h ) + h 4
and:

(2.10)

ˆΩ ∇R h (x ) 2 dx ≤ C h 2 E h (u h ) + h 4 .
The proof follows the lines of [START_REF] Friesecke | A hierarchy of plate models derived from nonlinear elasticity by Gamma-convergence[END_REF][START_REF] Lewicka | Scaling laws for non-Euclidean plates and the W 2,2 isometric immersions of Riemannian metrics[END_REF][START_REF] Lewicka | The Foppl-von Karman equations for plates with incompatible strains[END_REF], with necessary modifications in view of the expected error of the order h 4 . For completeness, we will present the details in the Appendix.

The lower bound

Theorem 3.1. Let u h ∈ W 1,2 (Ω h , R 3 ) be a sequence of deformations satisfying E h (u h ) ≤ Ch 4 .
Then there exists a sequence of translations c h ∈ R 3 and rotations Rh ∈ SO(3) such that the associated renormalizations:

(3.1) y h (x , x 3 ) = ( Rh ) t u h (x , hx 3 ) -c h ∈ W 1,2 (Ω 1 , R 3 )
have the following properties, where y 0 and b 0 are the unique solution to (1.7) (1.8). All convergences hold up to a subsequence:

(i) y h → y 0 in W 1,2 (Ω 1 , R 3 ) and 1 h ∂ 3 y h → b 0 in L 2 (Ω 1 , R 3 );
(ii) the scaled average displacements:

(3.2) V h (x ) = 1 h -1 2 -1 2 y h (x , x 3 ) -y 0 (x ) + hx 3 b 0 (x ) dx 3 converge in W 1,2 (Ω, R 3 ) to a limiting field V ∈ W 2,2
(Ω, R 3 ), satisfying the constraint:

(3.3) sym (∇y 0 ) t ∇V = 0;
(iii) the scaled tangential strains:

1 h sym (∇y 0 ) t ∇V h converge weakly in L 2 (Ω, R 2×2 ) to some S ∈ L 2 (Ω, R 2×2 sym ). (iv) Further, defining the quadratic forms Q 3 (F ) = D 2 W (Id 3 )(F, F ) and: (3.4) Q 2,A (x , F 2×2 ) = min Q 3 A(x ) -1 F A(x ) -1 ; F ∈ R 3×3 with F2×2 = F 2×2 ,
we have:

lim inf h→0 1 h 4 E h (u h ) ≥ I 4 (V, S) = 1 2 ˆΩ Q 2,A x , S + 1 2 (∇V ) t ∇V + 1 24 (∇ b 0 ) t ∇ b 0 dx + 1 24 ˆΩ Q 2,A x , (∇y 0 ) t ∇ p + (∇V ) t ∇ b 0 dx + 1 1440 ˆΩ Q 2,A x , (∇y 0 ) t ∇ d 0 + (∇ b 0 ) t ∇ b 0 dx , (3.5)
where the vector field p ∈ W 1,2 (Ω, R 3 ) is uniquely associated with V by:

(3.6) (∇y 0 ) t p = -(∇V ) t b 0 b 0 , p = 0. Proof. 1. Corollary 2.3 yields existence of R h ∈ W 1,2 (Ω, SO (3)
) such that (2.9) and (2.10) hold with Ch 4 and Ch 2 in their right hand sides, respectively. We rewrite these inequalities for the reader's convenience:

(3.7) 1 h ˆΩh ∇u h (x) -R h (x ) Q 0 (x ) + x 3 B 0 (x ) 2 dx ≤ Ch 4
and:

(3.8)

ˆΩ ∇R h (x ) 2 dx ≤ Ch 2 .
To prove the claimed convergence properties for (3.1), it is natural in view of (3.7) to set:

Rh = P SO(3) Ω h ∇u h (x)Q 0 (x ) -1 dx.
This projection is well defined, because for every x ∈ Ω, in view of (3.7):

dist 2 Ω h ∇u h Q -1 0 dx, SO(3) ≤ Ω h ∇u h Q -1 0 dx -R h (x ) 2 ≤ C Ω h (∇u h Q -1 0 -R h ) dx 2 + C Ω h R h dx -R h (x ) 2 ≤ C Ω h ∇u h -R h (Q 0 + x 3 B 0 ) Q -1 0 2 dx + C R h (x ) - Ω R h 2 ≤ C Ω h |∇u h -R h (Q 0 + x 3 B 0 )| 2 dx + C|R h (x ) - Ω R h | 2 ≤ Ch 4 + C|R h (x ) - Ω R h | 2
Now, taking the average on Ω, by the Poincaré-Wirtinger inequality and (3.8), we get:

dist 2 Ω h ∇u h Q -1 0 dx, SO(3) ≤ Ch 4 + C ˆΩ |∇R h | 2 ≤ Ch 2 ,
which proves that the average ffl Ω h ∇u h Q -1 0 dx is close to SO(3) and that:

(3.9) | Ω h ∇u h Q -1 0 dx -Rh | 2 ≤ Ch 2 .
Moreover:

Ω |R h -Rh | 2 dx = Ω h |R h -Rh | 2 dx ≤ C Ω h |R h - Ω R h | 2 + |( Ω R h ) - Ω h ∇u h Q -1 0 | 2 + Ω h | Rh - Ω h ∇u h Q -1 0 | 2 ≤ C Ω h |∇R h | 2 dx + C Ω h |∇u h -R h (Q 0 + x 3 B 0 )| 2 dx + Ch 2 ≤ Ch 2 , (3.10)
where the last estimate follows by (3.7), (3.8) and (3.9). Let now c h ∈ R 3 be such that ´Ω V h = 0 where V h is defined as in (3.2). Denote by ∇ h y h the matrix whose columns are given by ∂ 1 y h , ∂ 2 y h and ∂ 3 y h /h. Obviously:

(3.11) ∇ h y h (x , x 3 ) = ( Rh ) t ∇u h (x , hx 3 ).
Observe that:

ˆΩ1 |∇ h y h -Q 0 | 2 dx ≤ C Ω h |∇u h -Rh Q 0 | 2 dx ≤ C( Ω h |∇u h -R h (Q 0 + x 3 B 0 )| 2 dx + Ω h |x 3 R h B 0 | 2 dx + Ω h |R h -Rh | 2 dx) ≤ Ch 2
by (3.7) and (3.10). Therefore, ∇ h y h converges in L 2 (Ω 1 ) to Q 0 . Further, the sequence

{y h } is bounded in W 1,2 (Ω 1 )
, by the choice of c h . Passing to a subsequence we get that y h converges weakly in W 1,2 (Ω 1 ) and in view of the strong convergence of ∇y h we have:

y h → y 0 in W 1,2 (Ω 1 , R 3 ) and 1 h ∂ 3 y h → b 0 in L 2 (Ω 1 , R 3 ).
2. Note that, for every x ∈ Ω:

∇V h (x ) = 1 h 1/2 -1/2 ∇ h y h (x) -Q 0 (x ) dx 3 3×2 = 1 h 1/2 -1/2 ∇ h y h -( Rh ) t R h (Q 0 + hx 3 B 0 ) dx 3 3×2 + 1 h (( Rh ) t R h -Id 3 )Q 0 3×2 = I h 1 + I h 2 .
(3.12)

The first term above converges to 0. Indeed:

I h 1 2 L 2 (Ω) ≤ C h 2 Ω 1 |( Rh ) t ∇u h (x , hx 3 ) -( Rh ) t R h (Q 0 (x ) + hx 3 B 0 )| 2 dx ≤ C h 2 Ω h |∇u h (x , x 3 ) -R h (Q 0 + x 3 B 0 )| 2 dx ≤ Ch 2 .
(3.13)

Towards estimating the second term in (3.12), denote:

S h = 1 h (( Rh ) t R h -Id 3 ).
By (3.10) and (3.8), it follows that:

S h 2 L 2 (Ω) ≤ C h 2 ˆΩ |R h -Rh | 2 ≤ C and ∇S h 2 L 2 (Ω) ≤ C h 2 ˆΩ |∇R h | 2 ≤ C.
Passing to a subsequence, we can assume that:

(3.14) S h S weakly in W 1,2 (Ω),
which implies:

(3.15) I h 2 → (SQ 0 ) 3×2 in L 2 (Ω, R 3×2
). Consequently, by (3.12):

(3.16) ∇V h → (SQ 0 ) 3×2 in L 2 (Ω, R 3×2 ).
As before, we conclude that V h converges in W 1,2 (Ω) and that its limit V belongs to

W 2,2 (Ω, R 3 ), since ∇V = (SQ 0 ) 3×2 ∈ W 1,2 (
Ω). We now prove (3.3). By definition of S h :

(3.17) sym S h = - h 2 (S h ) t S h , so in view of the boundedness of {S h } in W 1,2 : sym S h L 2 (Ω) ≤ Ch S h 2 L 4 (Ω) ≤ Ch S h 2 W 1,2 (Ω) ≤ Ch. Consequently, S is a skew symmetric field. But (∇y 0 ) t ∇V = (Q t 0 SQ 0 ) 2×2 , hence (3.3) follows.
For future use, let us define p ∈ W 1,2 (Ω, R 3 ) by:

(3.18) [∇V | p] = SQ 0 . Since Q t 0 [∇V | p] ∈ so(3)
, it is easily checked that p is given solely in terms of V by:

(3.19) (∇y 0 ) t p = -(∇V ) t b 0 b 0 , p = 0.
3. We now want to establish convergence in (iii). In view of (3.12) we write:

1 h sym (Q t 0 ∇V h ) 2×2 (x ) = 1 h sym (Q t 0 I h 1 ) 2×2 + 1 h sym Q t 0 S h Q 0 2×2 = J h 1 + J h 2 .
(3.20)

We first deal with the sequence J h 2 . By (3.14), S h → S in L 4 (Ω) and so (3.17) implies:

1 h sym S h → - 1 2 S t S = 1 2 S 2 in L 2 (Ω).
Therefore:

(3.21) J h 2 → - 1 2 Q t 0 S t SQ 0 2×2 = - 1 2 (∇V ) t ∇V in L 2 (Ω).
We now prove that J h 1 converges. Recall that by (3.20), (3.12) and (3.11):

(3.22)

J h 1 = 1 h sym (Q t 0 I h 1 ) 2×2 = sym Q t 0 ( Rh ) t 1/2 -1/2 Z h (x , x 3 ) dx 3 2×2
where the rescaled strains Z h are defined by:

(3.23) Z h (x , x 3 ) = 1 h 2 ∇u h (x , hx 3 ) -R h (x )(Q 0 (x ) + hx 3 B 0 (x ))
. By (3.7), the sequence {Z h } is bounded in L 2 (Ω 1 , R 3 ). Therefore, up to a subsequence:

(3.24) Z h Z weakly in L 2 (Ω 1 , R 3 ).
It follows that:

(3.25) J h 1 J 1 := sym Q t 0 ( R) t 1/2 -1/2 Z(x , x 3 ) dx 3 2×2 weakly in L 2 (Ω).
which yields (iii) by (3.20) and (3.21).

4.

We now aim at giving the structure of the weak limit S of 1 h sym (Q t 0 ∇V h ) 2×2 in terms of the limiting fields V and Z. We have just seen that:

(3.26) S = J 1 - 1 2 (∇V ) t ∇V,
where J 1 is given by (3.25). As a tool, consider the difference quotients f s,h :

f s,h (x , x 3 ) = 1 h 2 s y h (x , x 3 + s) -y h (x , x 3 ) -hs b 0 + h x 3 + s 2 d 0 ,
and let us study for any s the convergence of f s,h as h → 0. In fact, we will show that f s,h p, weakly in W 1,2 (Ω 1 , R 3 ). Write:

f s,h (x , x 3 ) = 1 h 2 s 0 ∂ 3 y h (x , x 3 + t) -h( b 0 + h(x 3 + t) d 0 ) dt,
and observe that:

1 h 2 ∂ 3 y h -h( b 0 + hx 3 d 0 ) = 1 h ( Rh ) t ∇u h (x , hx 3 ) -(Q 0 + hx 3 B 0 ) e 3 = 1 h ( Rh ) t ∇u h (x , hx 3 ) -R h (Q 0 + hx 3 B 0 ) e 3 + S h (Q 0 + hx 3 B 0 )e 3 = h( Rh ) t Z h (x , x 3 )e 3 + S h (Q 0 + hx 3 B 0 )e 3 .
The first term in the right hand side above converges to 0 in L 2 (Ω 1 ) because {Z h } is bounded in L 2 (Ω 1 , R 3 ), while the second term converges to SQ 0 e 3 = S b 0 in L 2 (Ω 1 ) by (3.14). Note that SQ 0 e 3 = p by (3.18). Therefore, f s,h → p in L 2 (Ω 1 ). We now deal with the derivatives of the studied sequence. Firstly:

∂ 3 f s,h (x , x 3 ) = 1 s 1 h 2 ∂ 3 y h (x , x 3 + s) -h( b 0 + h(x 3 + s) d 0 ) - 1 h 2 ∂ 3 y h (x , x 3 ) -h( b 0 + hx 3 d 0 )
converges to 0 in L 2 (Ω 1 ). For i = 1, 2, the in-plane derivatives read as:

∂ i f s,h (x , x 3 ) = 1 h 2 s ( Rh ) t ∂ i u h (x , h(x 3 + s)) -( Rh ) t ∂ i u h (x , hx 3 ) -hs ∂ i b 0 + h x 3 + s 2 ∂ i d 0 = 1 s ( Rh ) t Z h (x , x 3 + s) -( Rh ) t Z h (x , x 3 ) e i + 1 h 2 s ( Rh ) t R h (Q 0 + h(x 3 + s)B 0 ) -( Rh ) t R h (Q 0 + hx 3 B 0 ) e i - 1 h B 0 e i + h x 3 + s 2 ∂ i d 0 .
The last two terms above can be written as: S h B 0 e i -x 3 + s 2 ∂ i d 0 , hence by (3.24):

∂ i f s,h (x , x 3 ) 1 s ( R) t Z(x , x 3 + s) -Z(x , x 3 ) e i + SB 0 e i -x 3 + s 2 ∂ i d 0 weakly in L 2 (Ω 1 , R 3 ),
where R ∈ SO(3) is an accumulation point of the rotations Rh . Consequently, f s,h p weakly in W 1,2 (Ω 1 , R 3 ) and, for i = 1, 2:

(3.27)

s∂ i p = ( R) t Z(x , x 3 + s) -Z(x , x 3 ) e i + sSB 0 e i -s x 3 + s 2 ∂ i d 0 ,
which proves that Z(x , •)e i has polynomial form and that:

(3.28) Rt Z(x , x 3 ) 3×2 = Rt Z(x , 0) 3×2 + x 3 (∇ p -(SB 0 ) 3×2 ) + x 2 3 2 ∇ d 0 .
By (3.24), it follows that:

J 1 = sym Q t 0 ( R) t Z(x , 0) 2×2 + 1 24 sym (Q t 0 ∇ d 0 ) 2×2 .
With (3.26), we finally arrive at the following identity that links S and V and Z:

S(x ) = sym Q t 0 ( R) t Z(x , 0) 2×2 + 1 24 sym (Q t 0 ∇ d 0 ) 2×2 - 1 2 (∇V ) t ∇V. (3.29)
5. We now prove the lower bound in (iv). Recall that by (3.23):

∇u h (x , hx 3 ) = R h (x )(Q 0 (x ) + hx 3 B 0 (x )) + h 2 Z h (x , x 3 ). Since Q 0 A -1 ∈ SO(3) we have: W (∇u h A -1 ) = W (Q 0 A -1 ) t (R h ) t ∇u h A -1 = W Id 3 + hJ + h 2 G h ,
where:

J (x , x 3 ) = x 3 A -1 (Q t 0 B 0 )A -1 (x ) ∈ so(3), G h (x , x 3 ) = A -1 Q t 0 (R h ) t Z h (x , x 3 )A -1 .

Note that by (3.24):

G h (x , x 3 ) G = A -1 Q t 0 ( Rt )Z(x , x 3 )A -1 weakly in L 2 (Ω 1 , R 3×3
). Define the "good sets": Ω h = {x ∈ Ω 1 ; h|G h | < 1}. By the above, the characteristic functions 1 Ω h converge to 1 in L 1 (Ω). Further, by frame invariance and Taylor expanding W on Ω h :

W Id 3 + hJ + h 2 G h = W e -hJ (Id 3 + hJ + h 2 G h ) = W (Id 3 + h 2 (G h - 1 2 J 2 ) + o(h 2 )) = 1 2 Q 3 h 2 (G h - 1 2 J 2 ) + o(h 4 ).
Therefore:

lim inf h→0 1 h 4 E h (u h ) ≥ lim inf h→0 1 h 4 ˆΩ1 1 Ω h W Id 3 + hJ + h 2 G h dx = lim inf h→0 1 2 ˆΩ1 Q 3 1 Ω h sym G h - 1 2 J 2 dx ≥ 1 2 ˆΩ1 Q 3 sym G - 1 2 J 2 dx, (3.30)
by the weak sequential lower semi-continuity of the quadratic form Q 3 in L 2 and in view of:

1 Ω h sym G h - 1 2 J 2 sym G - 1 2 J 2 weakly in L 2 (Ω 1 ).
Note that by (3.18) we have: Q t 0 SB 0 2×2 = -(∇V ) t ∇ b 0 and that:

J 2 = -J t J = -x 2 3 A -1 B t 0 B 0 A -1
. Therefore, using (3.28), the right hand side of (3.30) is bounded below by:

1 2 ˆΩ1 Q 2,A x , sym Q t 0 ( R) t Z(x , 0) + x 3 Q t 0 ∇ p + (∇V ) t ∇ b 0 + x 2 3 2 Q t 0 ∇ d 0 + (∇ b 0 ) t ∇ b 0 2×2 dx = 1 2 ˆΩ1 Q 2,A x , I(x ) + x 3 III(x ) + x 2 3 II(x ) dx.
Above we used (3.29) and we denoted:

I(x ) = S - 1 24 sym ((∇y 0 ) t ∇ d 0 ) + 1 2 (∇V ) t ∇V II(x ) = 1 2 sym ((∇y 0 ) t ∇ d 0 ) + 1 2 (∇ b 0 ) t ∇ b 0 III(x ) = sym((∇y 0 ) t ∇ p) + sym((∇V ) t ∇ b 0 ). (3.31)
Let L 2,A (x ) be the symmetric bilinear form generating the quadratic form Q 2,A (x ). Since the odd powers of x 3 integrate to 0 on the symmetric interval (-1/2, 1/2), we get:

ˆΩ1 Q 2,A x , I(x ) + x 3 III(x ) + x 2 3 II(x ) dx = ˆΩ Q 2,A (x , I(x )) dx + ( ˆ1/2 -1/2 x 2 3 dx 3 ) ˆΩ Q 2,A (x , III(x )) dx + ( ˆ1/2 -1/2 x 4 3 dx 3 ) ˆΩ Q 2,A (x , II(x )) dx + 2( ˆ1/2 -1/2 x 2 3 dx 3 ) ˆΩ L 2,A (x , I(x ), II(x )) dx = ˆΩ Q 2,A (x , I) + 1 12 ˆΩ Q 2,A (x , III) + 1 80 ˆΩ Q 2,A (x , II) + 2 12 ˆΩ L 2,A (x , I, II) dx = ˆΩ Q 2,A x , I + 1 12 II dx + 1 12 ˆΩ Q 2,A (x , III) dx + 1 180 ˆΩ Q 2,A (x , II) dx = I 4 (V, S),
by a direct calculation. This completes the proof of Theorem 3.1 in view of (3.30).

The upper bound

We now complete the proof of I 4 being the Γ-limit of h -4 E h , by proving that the lower bound (3.5) is optimal.

Theorem 4.1. Let V ∈ W 2,2 (Ω, R 3 ) and S ∈ L 2 (Ω, R 2×2 sym ) satisfy: sym (∇y 0 ) t ∇V = 0, S ∈ S := cl L 2 sym ((∇y 0 ) t ∇w); w ∈ W 1,2 (Ω, R 3 ) . (4.1)
Then there exists a sequence u h ∈ W 1,2 (Ω h , R 3 ) such that assertions (i), (ii) and (iii) of Theorem 3.1 are satisfied with R h = Id and c h = 0, and:

(4.2) lim sup h→0 1 h 4 E h (u h ) ≤ I 4 (V, S).
Proof. In the construction below, we will use the following notation. In view of (3.4), for every F 2×2 ∈ R 2×2 one can write:

Q 2,A (x , F 2×2 ) = min c∈R 3 Q 3 A -1 (F * 2×2 + sym(c ⊗ e 3 ))A -1 , (4.3)
where F * 2×2 denotes the R 3×3 matrix whose principal 2 × 2 minor equals F 2×2 . We will denote by c(x , F 2×2 ) the unique minimizer in (4.3). Note that c(x , •) is a linear function of F 2×2 and it depends only on its symmetric part (sym F 2×2 ).

1. Since S ∈ S, there exists a sequence w h ∈ W 1,2 (Ω, R 3 ) such that:

(4.4) sym (∇y 0 ) t ∇(w h + 1 24 d 0 ) → S in L 2 (Ω, R 2×2 )
and without loss of generality we can assume that each w h is smooth up to the boundary of Ω, together with:

(4.5) lim h→0 √ h w h W 2,∞ = 0.
Fix a small 0 ∈ (0, 1) and let v h ∈ W 2,∞ (Ω, R 3 ) be a sequence of Lipschitz deformations with the properties:

v h → V in W 2,2 (Ω, R 3 ), h v h W 2,∞ ≤ 0 , lim h→0 1 h 2 x ∈ Ω; v h (x ) = V (x ) = 0. (4.6)
We refer to [START_REF] Liu | A Lusin property of Sobolev functions[END_REF] and [START_REF] Friesecke | A hierarchy of plate models derived from nonlinear elasticity by Gamma-convergence[END_REF] for the construction of such truncated sequence v h . Define now p h ∈ W 1,∞ (Ω, R 3 ) by: (4.7)

p h = (Q t 0 ) -1 -(∇v h ) t b 0 0 ,
and also define the fields q h ∈ W 1,∞ (Ω, R 3 ), k 0 smooth and rh ∈ L ∞ (Ω, R 3 ) such that:

Q t 0 q h = 1 2 c x , 2(∇y 0 ) t ∇w h + (∇v h ) t ∇v h - (∇w h ) t b 0 0 - (∇v h ) t p h 1 2 | p h | 2 , Q t 0 k 0 = c x , (∇y 0 ) t ∇ d 0 + (∇ b 0 ) t ∇ b 0 - (∇ b 0 ) t d 0 | d 0 | 2 , Q t 0 rh = c x , (∇y 0 ) t ∇ p h + (∇v h ) t ∇ b 0 - (∇v h ) t d 0 p h , d 0 .
Finally, let r h ∈ W 1,∞ (Ω, R 3 ) be such that:

lim h→0 r h -rh L 2 = 0, lim h→0 √ h r h W 1,∞ = 0. (4.8)
It follows from the definition of the minimizing map c, that:

Q 3 A -1 2Q t 0 [∇w h | q h ] + [∇v h | p h ] t [∇v h | p h ] A -1 = Q 2,A x , 2(∇y 0 ) t ∇w h + (∇v h ) t ∇v h , Q 3 A -1 Q t 0 [∇ d 0 | k 0 ] + [∇ b 0 | d 0 ] t [∇ b 0 | d 0 ] A -1 = Q 2,A x , (∇y 0 ) t ∇ d 0 + (∇ b 0 ) t ∇ b 0 , Q 3 A -1 2Q t 0 [∇ p h | rh ] + 2[∇v h | p h ] t [∇ b 0 | d 0 ] t A -1 = Q 2,A x , (∇y 0 ) t ∇ p h + (∇v h ) t ∇ b 0 . (4.9)
Moreover, we have the following pointwise bounds:

| p h | ≤ C|∇v h |, |∇ p h | ≤ C(|∇v h | + |∇ 2 v h |), | q h | ≤ C(|∇w h | + |∇v h | 2 + |∇v h || p h | + | p h | 2 ) ≤ C(|∇w h | + |∇v h | 2 ), |∇ q h | ≤ C(|∇w h | + |∇ 2 w h | + |∇ 2 v h ||∇v h | + |∇v h | 2 ).
(4.10)

2. Consider the sequence u h ∈ W 1,∞ (Ω h , R 3 ) defined as:

u h (x , x 3 ) = y 0 (x ) + hv h (x ) + h 2 w h (x ) + x 3 b 0 (x ) + x 2 3 2 d 0 (x ) + x 3 3 6 k 0 (x ) + hx 3 p h (x ) + h 2 x 3 q h (x ) + hx 2 3 2 r h (x ).
For every (x , x 3 ) ∈ Ω 1 we write:

∇u h (x , hx 3 ) = Q 0 (x ) + Z h 1 (x , x 3 ) + Z h 2 (x , x 3 ),
where:

Z h 1 (x , x 3 ) = h[∇v h | p h ] + h 2 [∇w h | q h ] + hx 3 [∇ b 0 | d 0 ] + h 2 x 2 3 2 [∇ d 0 | k 0 ] + h 2 x 3 [∇ p h | r h ], Z h 2 (x , x 3 ) = h 3 x 3 3 6 [∇ k 0 | 0] + h 3 x 3 [∇ q h | 0] + h 3 x 3 2 [∇ r h | 0]. Since Q 0 A -1 ∈ SO(3)
, we get:

∇u h A -1 (x , hx 3 ) = Q 0 A -1 Id 3 + A -1 Q t 0 Z h 1 A -1 + A -1 Q t 0 Z h 2 A -1
and, in view of (4.6), (4.8) and (4.10), there follows for h sufficiently small:

A -1 Q t 0 Z h 1 A -1 + A -1 Q t 0 Z h 2 A -1 L ∞ ≤ C h ∇v h L ∞ + h p h L ∞ + h 2 ∇w h L ∞ + h 2 q h L ∞ + h ∇ b 0 L ∞ + h d 0 L ∞ + h 2 ∇ d 0 L ∞ + h 2 k 0 L ∞ + h 2 ∇ p h L ∞ + h 2 r h L ∞ + h 3 ∇ k 0 L ∞ + h 3 ∇ q h L ∞ + h 3 ∇ r h L ∞ ≤ C 0 .
By the left polar decomposition, there exists a further rotation R ∈ SO(3) such that:

R∇u h A -1 = (Id 3 + A -1 Q t 0 Z h 1 A -1 + A -1 Q t 0 Z h 2 A -1 ) t (Id 3 + A -1 Q t 0 Z h 1 A -1 + A -1 Q t 0 Z h 2 A -1 ) 1/2 = Id 3 + 2A -1 sym(Q t 0 Z h 1 )A -1 + A -1 (Z h 1 ) t Z h 1 A -1 + O(|Z h 2 |) 1/2 = Id 3 + A -1 sym(Q t 0 Z h 1 )A -1 + 1 2 A -1 (Z h 1 ) t Z h 1 A -1 + O | sym(Q t 0 Z h 1 ) + (Z h 1 ) t Z h 1 | 2 + O(|Z h 2 |).

Consider the set

:

Ω h = (x , x 3 ) ∈ Ω; v h (x ) = V (x ) .
Note that on Ω h we have: [START_REF] Barker | Existence and stability of viscoelastic shock profiles[END_REF]. Using Taylor's expansion, it follows that:

p h = p and Q t 0 [∇v h | p h ] ∈ so(
1 h 4 ˆΩh W ∇u(x , hx 3 )A -1 dx = 1 2h 4 ˆΩh Q 3 A -1 Q t 0 Z h 1 + 1 2 (Z h 1 ) t Z h 1 A -1 dx + E h 1 ,
where the error term E h 1 can be estimated by:

|E h 1 | ≤ C h 4 ˆΩh 2 sym(Q t 0 Z h 1 ) + (Z h 1 ) t Z h 1 3 + |Z h 2 | 2 + 2 sym(Q t 0 Z h 1 ) + (Z h 1 ) t Z h 1 |Z h 2 | dx.
Now on Ω h we also have, by (4.10):

2 sym(Q t 0 Z h 1 ) + (Z h 1 ) t Z h 1 ≤ C h 2 |∇w h | + h 2 |∇v h | 2 + h 2 + h 2 |∇v h | + h 2 |∇ 2 v h | + h 2 | r h | , |Z h 2 | ≤ Ch 3 1 + |∇ q h | + |∇ r h | ≤ Ch 3 1 + |∇w h | + |∇ 2 w h | + |∇ 2 v h ||∇v h | + |∇v h | 2 + |∇ r h | ,
and therefore, in view of (4.5), (4.8), (4.6) and V ∈ W 2,2 :

1 h 4 ˆΩh 2 sym(Q t 0 Z h 1 ) + (Z h 1 ) t Z h 1 3 dx ≤ C h 4 ˆΩh h 6 |∇w h | 3 + h 6 |∇v h | 6 + h 6 + h 6 |∇v h | 3 + h 6 |∇ 2 v h | 3 + h 6 | r h | 3 dx ≤ C h 4 h 2 ∇w h L ∞ (h 2 ∇w h L 2 ) 2 + h 6 ∇V 6 L 6 + h 6 |Ω| + h 6 ∇V 3 L 3 + h 6 ∇ 2 v h L ∞ ∇ 2 V 2 L 2 + ( √ h r h L ∞ ) 3 h 9/2 → 0 as h → 0.
Analogously:

1 h 4 ˆΩh |Z h 2 | 2 dx ≤ C h 4 ˆΩh h 5 + (h ∇v h L ∞ ) 2 h 4 |∇ 2 v h | 2 + h 6 |∇v h | 4 dx → 0 as h → 0, 1 h 4 ˆΩh 2 sym(Q t 0 )Z h 1 + (Z h 1 ) t Z h 1 |Z h 2 | dx ≤ C h 4 ˆΩh h 5 |∇w h | 2 + h 5 |∇ 2 w h | 2 + h 5 |∇v h | 2 + h 5 + h 5 |∇V | + h 5 |∇ 2 V | + h 5 | r h | + h 5 |∇V | 2 |∇ 2 V | + h 5 |∇V ||∇ 2 V | 2 dx ≤ C 0 .
We therefore conclude that:

(4.11) lim sup h→0 |E h 1 | ≤ C 0 .
4. Consider now the error due to integrating on the residual subdomain:

E h 2 = 1 h 4 ˆΩ1 \Ω h W ∇u h A -1 (x , hx 3 ) dx ≤ C h 4 ˆΩ1 \Ω h 2 sym(Q t 0 Z h 1 ) + (Z h 1 ) t Z h 1 2 + |Z h 2 | 2 dx.
Observe that, since the matrix field [∇v h | p h ] is Lipschitz, we have:

sym(Q t 0 [∇v h | p h ])(x ) ≤ C ∇v h W 1,∞ dist x , {v h = V } ≤ C 0 h dist x , {v h = V } → 0 in L ∞ (Ω).
The last inequality above follows by a standard argument by contradiction. If there was a sequence x h ∈ Ω such that dist(x h , {v h = V }) ≥ ch, this would imply that:

|{x ; v h (x ) = V (x )}| ≥ Ω ∩ B(x h , ch) ≥ ch 2 , contradicting (4.6)
. Consequently, by (4.5), (4.8), (4.6):

|E h 2 | ≤ C h 4 ˆΩ1 \Ω h h 2 sym(Q t 0 [∇v h | p h ]) dx + C h 4 ˆΩ1 \Ω h h 4 |∇w h | 2 + h 4 |∇v h | 4 + h 4 |∇ 2 v h | 2 + h 4 | r h | 2 + h 4 + h 6 |∇v h | 4 dx ≤ C h 4 o(h 2 )|Ω 1 \ Ω h | + C h 4 √ h ∇w h L ∞ h 7/2 |Ω 1 \ U h | 1/2 ∇w h L 2 + C|Ω 1 \ U h | ∇v h 4 L 8 + Ch ∇ 2 v h L ∞ 1 h ∇ 2 v h L 2 |Ω 1 \ U h | 1/2 + 1 h ( √ h r h L ∞ ) 2 |Ω 1 \ U h | + (h ∇ 2 v h L ∞ ) 2 ∇v h 2 L 4 |Ω 1 \ U h | 1/2 → 0 as h → 0.
Thus:

lim sup h→0 1 h 4 E h (u h ) ≤ lim sup h→0 1 h 4 ˆΩh 1 2 Q 3 A -1 sym(Q t 0 Z h 1 ) + 1 2 (Z h 1 ) t Z h 1 A -1 dx + C 0 .
Now on Ω h we have:

2 sym(Q t 0 Z h 1 ) + (Z h 1 ) t Z h 1 = 2h 2 sym(Q t 0 [∇w h | q h ]) + x 2 3 2 sym(Q t 0 [∇ d 0 | k 0 ]) + x 3 sym(Q t 0 [∇ p | r h ]) + h 2 [∇V | p] t [∇V | p] + x 2 3 [∇ b 0 | d 0 ] t [∇ b 0 | d 0 ] + 2x 3 sym([∇V | p] t [∇ b 0 | d 0 ]) + E h ,
where the present error E h is estimated by:

|E h | ≤ C h 3 |∇V ||∇w h | + h 3 |∇V | + h 3 |∇V ||∇ p| + h 3 |∇V || r h | + h 4 |∇w h | 2 + h 3 |∇w h | + h 4 |∇w h ||∇ p + h 4 |∇w h || r h | + h 3 + h 3 |∇ p| + h 3 | r h | + h 4 + h 4 |∇ p| + h 4 | r h | + h 4 |∇ p| 2 + h 4 | r h | 2 ≤ Ch 2 o(1) √ h|∇V | + 2 0 |∇ 2 V | + o(1) √ h + o(1) 0 √ h . (4.12)
Consequently:

lim sup h→0 1 h 4 E h (u h ) ≤ lim sup h→0 1 2 ˆΩh Q 3 A -1 sym(Q t 0 [∇w h | q h ]) + 1 2 x 2 3 sym(Q t 0 [∇ d 0 | k 0 ]) + x 3 sym(Q t 0 [∇ p | r h ]) + 1 2 [∇V | p] t [∇V | p] + 1 2 x 2 3 [∇ b 0 | d 0 ] t [∇ b 0 | d 0 ] + x 3 sym([∇V | p] t [∇ b 0 | d 0 ]) A -1 dx + C 0 = lim sup h→0 1 2 ˆΩh Q 3 A -1 sym(Q t 0 [∇w h | q h ]) + 1 2 [∇V | p] t [∇V | p] + 1 2 x 2 3 sym(Q t 0 [∇ d 0 | k 0 ]) + 1 2 x 2 3 [∇ b 0 | d 0 ] t [∇ b 0 | d 0 ] A -1 + Q 3 A -1 x 3 sym(Q t 0 [∇ p | r h ]) + x 3 sym([∇V | p] t [∇ b 0 | d 0 ]) A -1 dx + C 0 .
Denoting by:

I 1 (x ) = sym((∇y 0 ) t ∇w h ) + 1 2 (∇v h ) t ∇v h , I 2 (x ) = 1 2 sym((∇y 0 ) t ∇ d 0 ) + (∇ b 0 ) t ∇ b 0 ,
we have:

Q 3 A -1 I * 1 (x ) + sym(c(x , I 1 (x )) ⊗ e 3 ) + x 2 3 I * 2 (x ) + x 2 3 sym(c(x , I 2 (x )) ⊗ e 3 ) A -1 = Q 3 A -1 (I 1 (x ) + x 2 3 I 2 (x )) * + sym(c(x , I 1 (x ) + x 2 3 I 2 (x )) ⊗ e 3 ) A -1 = Q 2,A (I 1 (x ) + x 2 3 I 2 (x ) ,
where we have used the definition and linearity of the minimizing map c. Recalling the definitions of the curvature forms I(x ), II(x ) and III(x ) in (3.31), observe that I 2 (x ) = 2II(x ) and that 1 2 I 1 converges to I in L 2 by (4.4). Hence:

lim sup h→0 1 h 4 E h (u h ) ≤ 1 2 ˆΩ1 Q 2,A I(x ) + x 2 3 II(x ) dx + 1 2 ˆΩ1 Q 2,A x 3 III(x ) dx + C 0 = I 4 (V, S) + C 0 .
Since 0 > 0 was arbitrary, the proof is achieved by a diagonal argument.

5. Discussion of the von Kármán-like functional (3.5) Theorems 3.1 and 4.1 imply, as usual in the setting of Γ-convergence, convergence of almost-minimizers:

Corollary 5.1. If u h ∈ W 1,2 (Ω h , R 3 ) is a minimizing sequence to h -4 E h , that is: lim h→0 1 h 4 E h (u h ) -inf 1 h 4 E h = 0,
then the appropriate renormalizations y h = ( Rh ) t u h (x , hx 3 )-c h ∈ W 1,2 (Ω 1 , R 3 ) obey the convergence statements of Theorem 3.1 (i), (ii), (iii). The convergence of h -1 sym (∇y 0 ) t ∇V h to S in (iii) is strong in L 2 (Ω). Moreover, any limit (V, S) minimizes the functional I 4 .

Proof. The proof is standard. The only possibly nontrivial part is the strong convergence of the scaled tangential strains in (iii), which can be deduced as in Theorem 2.5 in [START_REF] Lewicka | Shell theories arising as low energy Gamma-limit of 3d nonlinear elasticity[END_REF].

Let us now compare the functional (3.5) with the von-Kármán theory of thin shells that has been derived in [START_REF] Lewicka | Shell theories arising as low energy Gamma-limit of 3d nonlinear elasticity[END_REF]. Recall that when S is a smooth 2d surface in R 3 , the Γ-limit of the scaled elastic energies h -4 1 h ´Sh W (∇u h ) on thin shells S h with mid-surface S, is:

(5.1) Ĩ4,S ( Ṽ , S) = 1 2 ˆS Q 2 S - 1 2 ( Ã2 ) tan dy + 1 24 ˆS Q 2 (∇( Ã N ) -ÃΠ) tan dy.
Above, Π stands for the shape operator of S and N is the unit normal vector to S. The subscript tan means taking the restriction of a quadratic form (or an operator) to the tangent space T y S. The arguments of Ĩ4,S are: (i) First order infinitesimal isometries Ṽ on S. These are vector fields Ṽ ∈ W 2,2 (S, R 3 ) with skew symmetric covariant derivative, so that one may define:

(5.2) Ã ∈ W 1,2 (S, so(3)) with Ã(y)τ = ∂ τ Ṽ (y) ∀y ∈ S ∀τ ∈ T y S;

(ii) Finite strains S on S. These are tensor fields S ∈ L 2 (S, R 2×2 sym ) such that:

(5.3) S = L 2 -lim h→0 sym(∇ wh ) tan for some wh ∈ W 1,2 (S, R 3 ).

In the present setting, denote S = y 0 (Ω) and observe that the 1-1 correspondence between Ṽ in (5.2) and V in (3.3) is given by the change of variables V = Ṽ • y 0 . The skew-symmetric tensor field à on T y S is then uniquely given by: Recall that the first of the two terms in the functional (5.1) measures the difference of order h 2 , between the (Euclidean) metric on S and the metric of the deformed surface. Indeed, the amount of stretching of S in the direction τ ∈ T y S, induced by the deformation u h = id + h Ṽ + h 2 w, has the expansion:

|∂ τ u h | 2 -|τ | 2 = h 2 2 ∂ τ w, τ + |∂ τ Ṽ | 2 + O(h 3 ) = 2h 2 (sym∇ w)τ, τ - 1 2 Ã2 τ, τ + O(h 3 ).
The leading order quantity in the right hand side above coincides with:

(sym∇w)e, e + 1 2 ∂ e V, ∂ e V = sym∇w + 1 2 (∇V ) t ∇V e, e ,
where we write τ = ∂ e y 0 , for any e ∈ R 2 . This is precisely the argument of the first term in I 4 (V, S), modulo the correction (∇ b 0 ) t ∇ b 0 (equal to the third fundamental form on S in case b 0 = N ), due to the incompatibility of the ambient Euclidean metric of S h with the given prestrain G on Ω h .

The second term in (5.1) measures the difference of order h, between the shape operator Π on S and the shape operator Π h on the deformed surface (id + h Ṽ )(S) whose unit normal we denote by N h . The amount of bending of S, in the direction τ ∈ T y S, induced by the deformation u h = id + h Ṽ can be estimated by [START_REF] Lewicka | Shell theories arising as low energy Gamma-limit of 3d nonlinear elasticity[END_REF]:

(Id + h Ã) -1 Π h (Id + h Ã)τ -Πτ = (Id + h Ã) -1 ∂ τ N h + O(h 2 ) τ -Πτ = (Id + h Ã) -1 (Id + h Ã)Πτ + h(∂ τ A) N + O(h 2 ) -Πτ = (Id -h Ã)h(∂ τ Ã) N + O(h 2 ) = h(∂ τ Ã) N + O(h 2 ) = h ∇( Ã N ) -ÃΠ + O(h 2 ).
The leading order term in this expansion coincides with the term (∇y 0 ) t ∇ p+(∇V ) t ∇ b 0 when b 0 = N , because in view of (5. where we again wrote τ = ∂ e y 0 ∈ T y 0 (x ) S, for any e ∈ R 2 . This is precisely the argument in the second term in I 4 (V, S).

In the next section we identify the geometric significance of the last term in (3.5).

The scaling optimality

In this section, we prove the following crucial result: Theorem 6.1. Assume (1.5), together with:

(6.1) sym (∇y 0 ) t ∇ d 0 + (∇ b 0 ) t ∇ b 0 = 0,
where y 0 , b 0 and d 0 are defined in (1.7), (1.8), (1.10). Then the metric G is flat, i.e. Riem(G) ≡ 0 in Ω h . Equivalently: min E h = 0 for all h.

Observe that when b 0 = N , then by (1.10) there must be d 0 = 0, and hence condition (6.1) becomes: N ≡ const. This is consistent with our previous observation that when Ge 3 = e 3 , then already condition (1.7) is enough to conclude immersability of G in R 3 . Equivalently, G 2×2 is immersible in R 2 , so that indeed y 0 (Ω) must be planar in this case.

Towards a proof of Theorem 6.1, recall that Riem(G) is the covariant Riemann curvature tensor, whose components R .... and their relation to the contravariant curvatures in R .

... are:

R iklm = 1 2 ∂ kl G im + ∂ im G kl -∂ km G il -∂ il G km + G np Γ n kl Γ p im -Γ n km Γ p il R iklm = G is R s klm ,
where we used the Einstein summation convention and the Christoffel symbols:

(6.2) Γ n kl = 1 2 G ns ∂ k G sl + ∂ l G sk -∂ s G kl .
In view of the symmetries in Riem(G) of a 3-dimensional metric G, its flatness is equivalent to the vanishing of the following curvatures:

R 1212 , R 1213 , R 1223 , R 1313 , R 1323 , R 2323 .
The proof of Theorem 6.1 is a consequence of the following observation.

Theorem 6.2. Assume (1.5) and let y 0 , b 0 and d 0 be defined as in (1.7), (1.10). Then:

(6.3) sym (∇y 0 ) t ∇ d 0 + (∇ b 0 ) t ∇ b 0 = R 1313 R 1323 R 1323 R 2323 .
Proof. 1. We have:

R 1313 = - 1 2 ∂ 11 G 33 + G np Γ n 13 Γ p 13 -Γ n 11 Γ p 33 , R 2323 = - 1 2 ∂ 22 G 33 + G np Γ n 23 Γ p 23 -Γ n 22 Γ p 33 , R 1323 = - 1 2 ∂ 12 G 33 + G np Γ n 13 Γ p 23 -Γ n 12 Γ p 33 .
On the other hand, in view of (1.10):

∀i, j = 1, 2 1 2 ∂ i y 0 , ∂ j d 0 + ∂ j y 0 , ∂ i d 0 = 1 2 ∂ j ∂ i y 0 , d 0 + ∂ i ∂ j y 0 , d 0 -∂ ij y 0 , d 0 = - 1 2 ∂ ij G 33 -∂ ij y 0 , d 0 because: ∂ j ∂ i y 0 , d 0 + ∂ i ∂ j y 0 , d 0 = -∂ ij | b 0 | 2 = -∂ ij G 33 .
Consequently, the formula (6.3) will follow, if we establish:

(6.4) ∀i, j = 1, 2 ∂ ij y 0 , d 0 = G np Γ n ij Γ p 33 and ∂ i b 0 , ∂ j b 0 = G np Γ n i3 Γ p j3 .
2. Before proving (6.4) we gather some useful formulas. Note that

∂ i G = 2 sym ((∂ i Q) t Q) for i = 1, 2. Therefore, by direct inspection: (6.5) ∀i, j, k = 1, 2 ∂ ij y 0 , ∂ k y 0 = 1 2 (∂ i G kj + ∂ j G ki -∂ k G ij ).
Also, recall that condition (1.7) is equivalent to (see [START_REF] Bhattacharya | Plates with incompatible prestrain[END_REF], proof of Theorem 5.3, formula (5.8)):

(6.6) ∀i, j = 1, 2 ∂ ij y 0 , b 0 = 1 2 (∂ i G j3 + ∂ j G i3 ).
Therefore, for all i, j = 1, 2:

∂ j y 0 , ∂ i b 0 = ∂ i ∂ j y 0 , b 0 -∂ ij y 0 , b 0 = 1 2 (∂ i G j3 -∂ j G i3 ), ∂ i b 0 , b 0 = 1 2 ∂ i G 33 . (6.7)
We now express ∂ ij y 0 , ∂ i b 0 and d 0 in the basis {∂ 1 y 0 , ∂ 2 y 0 , b 0 }, writing:

∂ ij y 0 = α 1 ij ∂ 1 y 0 + α 2 ij ∂ 2 y 0 + α 3 ij b 0 , ∂ i b 0 = β 1 i ∂ 1 y 0 + β 2 i ∂ 2 y 0 + β 3 i b 0 , d 0 = γ 1 ∂ 1 y 0 + γ 2 ∂ 2 y 0 + γ 3 b 0 .
(6.8) By (6.5), (6.6), (6.7) and (1.10), it follows that:

G α 1 ij , α 2 ij , α 3 ij t = GQ -1 0 ∂ ij y 0 = Q t 0 ∂ ij y 0 = 1 2 ∂ i G 1j + ∂ j G 1i -∂ 1 G ij , ∂ i G 2j + ∂ j G 2i -∂ 2 G ij , ∂ i G 3j + ∂ j G 3i , G β 1 i , β 2 i , β 3 i t = GQ -1 0 ∂ i b 0 = Q t 0 ∂ b 0 = Q t 0 ∂ i b 0 = 1 2 ∂ i G 13 -∂ 1 G i3 , ∂ i G 23 -∂ 2 G i3 , ∂ i G 33 t , G γ 1 , γ 2 , γ 3 t = GQ -1 0 d 0 = Q t 0 d 0 = - 1 2 ∂ 1 G 33 , ∂ 2 G 33 , 0 t .
In view of (6.2) we then obtain, for all i, j = 1, 2:

(α 1 ij , α 2 ij , α 3 ij ) = (Γ 1 ij , Γ 2 ij , Γ 3 ij ), (β 1 i , β 2 i , β 3 i ) = (Γ 1 i3 , Γ 2 i3 , Γ 3 i3 ), (γ 1 , γ 2 , γ 3 ) t = (Γ 1 33 , Γ 2 33 , Γ 3 33 )
, so that (6.8) becomes:

∂ ij y 0 = Γ 1 ij ∂ 1 y 0 + Γ 2 ij ∂ 2 y 0 + Γ 3 ij b 0 , ∂ i b 0 = Γ 1 i3 ∂ 1 y 0 + Γ 2 i3 ∂ 2 y 0 + Γ 3 i3 b 0 , d 0 = Γ 1 33 ∂ 1 y 0 + Γ 2 33 ∂ 2 y 0 + Γ 3 33 b 0 .
(6.9)

3. We now prove (6.4). Keeping in mind that Q T 0 Q 0 = G, the scalar products of expressions in (6.9) are:

∂ ij y 0 , d 0 = Γ n ij ∂ n y 0 + Γ 3 ij b 0 , Γ p 33 ∂ p y 0 + Γ 3 33 b 0 = G np Γ n ij Γ p 33 , ∂ i b 0 , ∂ j b 0 = Γ n i3 ∂ n y 0 + Γ 3 i3 b 0 , Γ p j3 ∂ p y 0 + Γ 3 j3 b 0 = G np Γ n i3 Γ p j3
. exactly as claimed in (6.4). This ends the proof of Theorem 6.2 and also of Theorem 6.1.

Two examples

In this section we compute the energy I 4 (V, S) in the two particular cases of interest:

G(x , x 3 ) = diag(1, 1, λ(x )) and G(x , x 3 ) = λ(x )Id 3 .
Let p be as in the definition (3.6). Writing:

p = α 1 ∂ 1 y 0 + α 2 ∂ 2 y 0 + α 3 b 0 , we obtain: G α 1 , α 2 , α 3 t = -∂ 1 V , b 0 , ∂ 2 V , b 0 , 0 t . Consequently: p = -G 1i ∂ i V , b 0 ∂ 1 y 0 -G 2i ∂ i V , b 0 ∂ 2 y 0 -G 3i ∂ i V , b 0 b 0 . (7.1)
Lemma 7.1. Let λ : Ω → R be smooth and strictly positive. Consider the metric of the form: G(x , x 3 ) = diag(1, 1, λ(x )). Then:

(i) G is immersible in R 3 if and only if:

M λ = ∇ 2 λ - 1 2λ ∇λ ⊗ ∇λ ≡ 0 in Ω, while the condition M λ ≡ 0 is equivalent to: ch 4 ≤ inf E h ≤ Ch 4 . (ii) The Γ-limit energy functional I 4 in (3.5) becomes: ∀w ∈ W 1,2 (Ω, R 2 ) ∀v ∈ W 2,2 (Ω, R) I 4 (v, w) = 1 2 ˆΩ Q 2 sym∇w + 1 2 ∇v ⊗ ∇v + 1 96λ ∇λ ⊗ ∇λ dx + 1 24 ˆΩ Q 2 √ λ∇ 2 v + 1 5760 ˆΩ Q 2 M λ dx ,
where Q 2 is independent of x and it is defined by Q 2,Id in (3.4).

Proof. Part (i) of the assertion has been shown in [START_REF] Bhattacharya | Plates with incompatible prestrain[END_REF]. For (ii), note first that:

y 0 (x ) = x and Q 0 = A = diag(1, 1, √ λ).
Consequently, directly from (3.4) we see that Q 2,A = Q 2,Id , which we denote simply by Q 2 . Further, in view of (4.1), every admissible limiting strain S ∈ S has the form S = sym∇w for some w ∈ W 1,2 (Ω, R 2 ). Also, without loss of generality, every admissible limiting displacement V is of the form: V = (0, 0, v) for some v ∈ W 2,2 (Ω, R). We now compute, using (1.9), (6.9) and (7.1):

b 0 = √ λe 3 , d 0 = - 1 2 (∂ 1 λ, ∂ 2 λ, 0), p = - √ λ(∂ 1 v, ∂ 2 v, 0). Therefore: (∇ b 0 ) t ∇ b 0 = 1 4λ ∇λ ⊗ ∇λ, ( ∇y
0 ) t ∇ d 0 = - 1 2 ∇ 2 λ, ( ∇y
0 ) t ∇ p = - 1 2 √ λ ∇v ⊗ ∇λ - √ λ∇ 2 v, (∇V ) t ∇ b 0 = 1 2 √ λ ∇v ⊗ ∇λ.
This ends the proof of Lemma 7.1 in view of (3.5).

Lemma 7.2. Let λ : Ω → R be smooth and strictly positive. Consider the metric G(x , x 3 ) = λ(x )Id 3 . Denote f = 1 2 log λ. Then: (i) Condition (1.7) is equivalent to ∆f = 0, which is also equivalent to the immersability of the metric G 2×2 in R 2 . (ii) Under condition (1.7), condition (6.1) can be directly seen as equivalent to Ric(G) = 0 and therefore to the immersability of G.

(iii) The Γ-limit energy functional in (3.5) has the following form:

I 4 (V, S) = 1 2 ˆΩ e -2f Q 2 S + 1 2 (∇V ) t ∇V + 1 24 e 2f ∇f ⊗ ∇f dx + 1 24 ˆΩ Q 2 2∇V 3 ⊗ ∇f -∇ 2 V 3 -∇V 3 , ∇f Id 2 dx + 1 1440 ˆΩ Q 2 e f Ric(G) 2×2 dx ,
where Q 2 is as in Lemma 7.1, and where Ric(G) 2×2 denotes the tangential part of the Ricci curvature tensor of G, i.e.:

Ric(G) 2×2 = R 11 R 12 R 12 R 22 .
Proof. The part (i) has been deduced in [START_REF] Bhattacharya | Plates with incompatible prestrain[END_REF], together with the expression:

(7.2) Ric(G) = -(∇ 2 f -∇f ⊗ ∇f ) * -(∆f + |∇f | 2 )Id 3 .
We now consider the case when (1.7) holds. By (i) the metric G 2×2 is immersible in R 2 and in particular N = e 3 . Writing V = (V 1 , V 2 , V 3 ), from (1.9), (6.9) and (7.1) we obtain: In the same manner, we arrive at:

b 0 = √ λe 3 , d 0 = -∂ 1 f ∂ 1 y 0 + ∂ 2 f ∂ 2 y 0 , p = - 1 √ λ ∂ 1 V 3 ∂ 1 y 0 + ∂ 2 V 3 ∂ 2 y 0 . (∇ b 0 ) t ∇ b 0 = e 2f ∇f ⊗ ∇f, (
1 λ ∂ 2 y 0 , ∂ 2 d 0 = -(∂ 22 f + |∇f | 2 ), 1 λ ∂ 2 y 0 , ∂ 1 d 0 = -∂ 12 f, 1 λ ∂ 1 y 0 , ∂ 2 d 0 = -∂ 21 f.
Consequently, (∇y 0 ) t ∇ d 0 is already a symmetric matrix, and:

(∇y 0 ) t ∇ d 0 = -e 2f (∇ 2 f + |∇f | 2 Id 2 ).
In particular, under condition ∆f = 0, the formula (7.2) yields:

sym (∇y 0 ) t ∇ d 0 + (∇ b 0 ) t ∇ b 0 = e 2f Ric(G) 2×2 ,
which we directly see to be equivalent with ∇f = 0 and hence with Ric(G) = 0. This establishes (ii). We now compute the remaining quantities appearing in the expression of I 4 . Firstly:

∇ p = 1 2λ 3/2 ∇y 0 (∇V 3 ⊗ ∇λ) -

1 √ λ ∇y 0 ∇ 2 V 3 - 1 √ λ ∂ 1 V 3 (∂ 11 y 0 , ∂ 12 y 0 ) + ∂ 2 V 3 (∂ 12 y 0 , ∂ 22 y 0 ) .
Using the relations between ∂ ij y 0 , ∂ k y 0 and ∂ l G in (6.5), we obtain: In a similar manner, it follows that:

(∇y 0 ) t ∇ p = 1 2λ 3/2 G 2×2 ∇V 3 ⊗ ∇λ - 1 √ λ G 2×2 ∇ 2 V 3 - 1 2 √ λ ∇V 3 , ∇λ ∇V 3 , ∇λ ⊥ -∇V 3 , ∇λ ⊥ ∇V 3 , ∇λ ,
sym(∇y 0 ) t ∇ d 0 = -λ ∇ 2 f + |∇f | 2 Id 2 .
Since Q 2,A (x ) = λ -1 Q 2 , the formula in (3.5) becomes: 1. For every x ∈ Ω denote D x ,δ = B(x , δ) ∩ Ω and B x ,δ,h = D x ,δ × (-h/2, h/2). For short, we write B x ,2h = B x ,2h,h and B x ,h = B x ,h,h . Apply Lemma 2.2 to the set V h = B x ,2h to get a rotation R x ,2h ∈ SO(3) such that, with a universal constant C:

I 4 (V, S) = 1 2 ˆΩ e -2f Q 2 S
1 h ˆBx ,2h
∇u h (z) -R x ,2h Q 0 (z ) + z 3 B 0 (z )

( 5

 5 .4) Ã(y 0 (x ))∂ e y 0 = ∂ e V (x ) and à b 0 = p ∀e ∈ R 2 ,and the finite strains in (5.3) are related to (4.1) by: S(y 0 (x ))∂ e y 0 , ∂ e y 0 = S(x )e, e ∀e ∈ R 2 .

  4): (∂ τ Ã) b 0 , τ = (∂ e ( à b 0 ), ∂ e y 0 -( Ã∂ e b 0 , ∂ e y 0 = (∂ e p, ∂ e y 0 + (∂ e b 0 , Ã∂ e y 0 = (∇y 0 ) t ∇ p e, e -(∇V ) t ∇ b 0 e, e ,

λ λ∂ 11 f + 1 2 ∂ 1 λ∂ 1 f + 1 2 ∂ 2

 222 ∇V ) t ∇ b 0 = e f ∇V 3 ⊗ ∇f.Further, observe that:∂ i d 0 = -(∂ 1i f ∂ 1 y 0 + ∂ 2i f ∂ 2 y 0 + ∂ 1 f ∂ 1i y 0 + ∂ 2 f ∂ 2i y 0 ), and so: 1 λ ∂ 1 y 0 , ∂ 1 d 0 = -1 λ∂ 2 f = -(∂ 11 f + |∇f | 2 ).

  0 ) t ∇ p = √ λ sym ∇V 3 ⊗ ∇f -√ λ∇ 2 V 3 -√ λ ∇V 3 , ∇λ Id 2 .

2 dz≤( 8 . 1 ) 2 =dy 2 ≤( 8 . 4 )dy 2 = 2 ≤

 281228422 C E (u h , B x ,2h ) + h 3 B x ,2h .Consider a family of mollifiers η x ∈ C ∞ (Ω, R), parametrized by x ∈ Ω:ˆΩ η x = 1 h , η x L ∞ (Ω) ≤ C h 3 , ∇ x η x L ∞ (Ω) ≤ C h 4 and (supp η x ) ∩ Ω ⊂ D x ,h . Define Rh ∈ W 1,2 (Ω, R 3×3 ) as: (8.2) Rh (x ) = ˆΩh η x (z )∇u h (z) Q 0 (z ) + z 3 B 0 (z ) -1 dz.We then have:1 h ˆBx ,h |∇u h (z)-Rh (z ) Q 0 (z ) + z 3 B 0 (z ) | 2 dz ≤ C h ˆBx ,2h ∇u h (z) -R x ,2h Q 0 (z ) + z 3 B 0 (z ) h | Rh (z ) -R x ,2h | 2 |Q 0 (z ) + z 3 B 0 (z )| 2 dz ≤ C E h (u h , B x ,2h ) + h 3 B x ,2h + C h ˆBx ,h | Rh (z ) -R x ,2h | 2 dz,(8.3)where we have used (8.1) and Q 0 (z ) + z 3 B 0 (z ) L ∞ ≤ C. Now, for every z ∈ B x ,h we have:| Rh (z ) -R x ,2h | 2 = ˆΩh η z (y )∇u h (y) Q 0 (y ) + y 3 B 0 (y ) -1 dy -R x ,2h ˆΩh η z (y ) ∇u h (y) -R x ,2h Q 0 (y ) + y 3 B 0 (y ) Q 0 (z ) + y 3 B 0 (z ) -1 C ˆBz ,h η z (y ) 2 dy ˆBz ,h ∇u h (y) -R x ,2h Q 0 (y ) + y 3 B 0 (y ) ∇u h (y) -R x ,2h Q 0 (y ) + y 3 B 0 (y ) 2 dy ≤ C h 2 E h (u h , B x ,2h ) + h 3 B x ,2h .In a similar way, in view of ´Ωh ∇ z η z (y ) dy = 0, it follows that:|∇ Rh (z )| 2 = ˆΩh ∇ z η z (y )∇u h (y) Q 0 (y ) + y 3 B 0 (y ) -1 ˆBx ,2h ∇ z η z (y ) ∇u h (y) Q 0 (y ) + y 3 B 0 (y ) -1 -R x ,2h dy C ˆΩh ∇ z η z (y )

  -2f Q 2 2e f ∇V 3 ⊗ ∇f -e f ∇ 2 V 3 -e f ∇V 3 , ∇f Id 2 dx + 1 1440 ˆΩ e -2f Q 2 e 2f Ric(G) 2×2 dx ,

	+ ˆΩ e (7.3) 1 24	+	1 2	(∇V ) t ∇V +	1 24	e 2f ∇f ⊗ ∇f dx
	which implies the result.

8. Appendix: a proof of Corollary 2.3

  2 dy ˆBx ,2h ∇u h (y) -R x ,2h Q 0 (y ) + y 3 B 0 (y ) E h (u h , B x ,2h ) + h 3 B x ,2h . From (8.4) we obtain:ˆBx ,h | Rh (z ) -R x ,2h | 2 dz ≤ C h 2 ˆBx ,h E h (u h , B x ,2h ) + h 4 |B x ,2h | dz ≤ Ch E h (u h , B x ,2h ) + h 3 |B x ,2h | ,and therefore by (8.3) we further see that:1 h ˆBx ,h |∇u h (z)-Rh (z ) Q 0 (z ) + z 3 B 0 (z ) | 2 dz ≤ C E h (u h , B x ,2h ) + h 3 |B x ,2h | .Covering Ω h by a finite family of sets {B x ,h }, such that the intersection number of the doubled covering {B x ,2h } is independent of h, applying (8.5) and summing over the covering, it follows that:1 h ˆΩh |∇u h (z) -Rh (z ) Q 0 (z ) + z 3 B 0 (z ) | 2 dz ≤ C E h (u h ) + h 4 . E h (u h , B x ,2h ) + h 3 |B x ,2h | dz ≤ C h 2 E h (u h , B x ,2h ) + h 3 |B x ,2h| , and by the same covering argument:ˆΩh |∇ Rh (z )| 2 dz ≤ C h 2 E h (u h ) + h 4 .3. Note that, in the above two estimates, we can replace Rh byR h = P SO(3) Rh ∈ W 1,2 (Ω, SO(3)).Firstly, the projection in question is well defined in view of (8.4), since:dist 2 Rh , SO(3) ≤ | Rh -R x ,2h | ≤ C h 2 E h (u h ) + h 4, which is small because of the hypothesis α < 2. Moreover:1 h ˆBx ,h |∇u h (z) -R h (z ) Q 0 (z ) + z 3 B 0 (z ) | 2 dz ≤ C h ˆBx ,h ∇u h (z) -Rh (z ) Q 0 (z ) + z 3 B 0 (z ) Rh (z ) -R h (z )| 2 |Q 0 (z ) + z 3 B 0 (z )| 2 dz ≤ C E h (u h , B x ,2h ) + h 3 |B x ,2h |because of (8.5) and(8.4). Finally, the previous covering argument clearly implies (2.9), and ´Ω |∇R h | 2 dz ≤ C ´Ω |∇ Rh | 2 dz yields (2.10).

	≤ h 4 (8.5) C ˆDx ,h 2. In a similar fashion we obtain: |∇ Rh (z )| 2 dz ≤ + C C h 4 h ˆBx ,h ˆDx ,h |	2	2 dz	dy