Relative Bending Energy for Weakly Prestrained Shells
Silvia Jimenez Bolanos Anna Zemlaynova


In this paper, we derive a dimensionally reduced model for a thin film prestrained with a given incompatible Riemannian metric: 

    G^h(x',x_3)=I_3+2h^γ S(x')+2h^γ/2 x_3B(x')+h.o.t,    γ>2,
 
where 0<h≪ 1 is the thickness of the film.  The problem is studied rigorously by using a variational approach and establishing the Γ-convergence of the non-Euclidean version of the nonlinear elasticity functional.  It is shown that the residual nonlinear elastic energy scales as O(h^γ+2) as h→ 0. 


A weakly supervised adaptive triplet loss for deep metric learning
    Xiaonan Zhao1, Huan Qi1, Rui Luo1 and Larry Davis1,2



1Amazon.com Services, Inc., Seattle, USA 2University of Maryland, College Park, USA

{xiaonzha,qihuan,luorui,lrrydav}@amazon.com

    December 30, 2023
=============================================================================================================================================================================================





§ INTRODUCTION.


The present study is concerned with the derivation of dimensionally reduced models for limiting behavior of thin films prestrained with a family of incompatible metrics G^h.  The motivation behind this study comes from applications for thin objects with internal prestrain such as growing tissues and various manufactured phenomena (for instance, polymer gels, atomically thin graphene layers, and plastically strained sheets).  Shape formation driven by internal prestrain is a very active area of research which has been tackled before by various  analytic and numerical arguments; see for instance <cit.>. 

The dimension reduction problems for thin plates and shells consist of minimizing a nonlinear elastic energy functional representing a mismatch between the deformation of the film and the target Riemannian metric G.  In <cit.>, the authors considered the case G=_3 and derived nonlinear membrane models, for planar membranes and shells, from the variational formulation of the  three-dimensional nonlinear elasticity.  They showed that the deformations that minimize, or almost minimize, the total energy weakly converge in a Sobolev space towards deformations that minimize a nonlinear membrane energy, as the thickness of the body goes to zero.  The limiting nonlinear membrane energy was obtained by Γ-convergence of the sequence of three-dimensional energies.  The notion of Γ-convergence, introduced by Ennio De Giorgi in a series of papers published between 1975 and 1983 <cit.>, has become the standard notion of convergence for variational problems <cit.>.

In the fundamental papers <cit.>, the authors derived the hierarchy of limiting theories of thin plates which arise as Γ- limits of three dimensional nonlinear theory in the classical elasticity.  In particular, they were able to show that von Kármán equations can be obtained rigorously from variational formulation of the nonlinear elasticity. The limiting theories differ in their scaling as powers of the thickness h of the plate or shell depending on the scaling of the applied external forces.

The work in this area has been further extended to limiting theories for thin shells in <cit.>.  The first results for thin nonlinear elastic shells were obtained in <cit.> for the scaling β=0.  The case  β=2  was investigated in <cit.>.  The nonlinear theory for shells with varying thickness with external loading was studied in <cit.>.  In <cit.>, the authors studied higher order β≥ 4 energy scaling for thin nonlinear elastic shells.  The paper <cit.> further extends the results of <cit.>, by proving that the equilibria of nonlinear energy functional for a thin shell converge to the equilibria of the von Kármán functional.  The intermediate case of 2<β<4 was studied in 
<cit.>.  Finally, the results were summarized and the conjecture about the infinite hierarchy of limiting two-dimensional models was made in <cit.>.  The problem for thin shells has been revisited in <cit.> to study the Γ-limit for thin shells with certain scaling of the applied forces.  In the papers mentioned above, the deformation of the plate or shell has been activated due to an application of external forces and not due to the prestrain.


In the context of the prestrain-driven response, the parallel theories are differentiated by the embeddability properties of the target metrics and, a-posteriori, by the emergence of isometry constraints on deformations with low regularity.  In turn, results on thin limit models have ramifications for the three dimensional original model with regard to energy scaling laws, understanding of the role of curvature in identifying the material's mechanical properties, and in the consequences of the symmetry and the symmetry breaking in the solutions to the resulting Euler-Lagrange equations.  The first work to rigorously study non-trivial configurations of thin prestrained flat films was produced in <cit.>. In this paper, the variational formulation of the problem was introduced by using a nonlinear elasticity functional and the necessary and sufficient conditions for existence of a W^2,2 isometric immersion were obtained.  The targeted metric in <cit.> does not depend on the thickness variable of the plate. Further results for quantization of the elastic energy for the case of thickness-independent metric have been obtained in <cit.> and <cit.>.  A survey of available models for the thickness-independent prestrain is given in <cit.>. The question of immersability for a thickness-dependent Riemann metric satisfying certain general conditions has been decisively answered in <cit.>.  In <cit.>, the dimension reduction for oscillatory metrics was obtained.  The prescribed incompatibility metric in this paper exhibits a nonlinear dependence on the transversal variable. Asymptotic theories for shallow shells prestrained with a growth tensor linearly dependent on the transversal variable were studied in <cit.>. A survey of results available in prestrained elasticity can be found in <cit.>. 

In our paper, as in <cit.>, the prestrain metrics G^h (see (<ref>)) are perturbations of the flat I_3 metric.  In particular, in <cit.> the authors derived a new variational model consisting of minimizing a biharmonic energy of the out-of plane displacement v∈ W^2,2(Ω,ℝ), satisfying the Monge-Ampère constraint ∇^2v=f, where f=- curl^T curl S_2×2 is the linearized Gauss curvature of the Riemannian metrics in (<ref>).  The work in <cit.> was done in the parameter range 0<γ<2, whereas the case γ=2 was treated in <cit.>, leading to the derivation of the Föppl-von Kármán equations accounting for the presence of the prestrain.  The main contribution of our paper is to develop the analysis for the parameter range γ>2.   We identified  the asymptotic behavior of the minimizers of I_W^h(u^h) as h→0 (see (<ref>)), through deriving the Γ-limit of the rescaled energies 1/h^γ+2I_W^h(u^h).  These new outcomes are presented in Theorem <ref> and Theorem <ref>. A generalization of the current results to the growth tensors A^h(x',x_3)=I_3+h^αS(x')+h^γ/2x_3B(x'), with arbitrary powers α and γ, is currently in preparation <cit.>. 

This paper is organized as follows.  In Section <ref>, we give the problem formulation.  In Section <ref>, we present the main results of this paper, which are stated in Theorem <ref> and Theorem <ref>.  Finally, in Section <ref> and Section <ref>, we present the proofs of Theorem <ref> and Theorem <ref>, respectively.



§ PROBLEM FORMULATION: NON-EUCLIDEAN ELASTICITY MODEL.


Let the smooth invertible growth tensors A^h=[A_ij^h]:Ω^h→ℝ^3× 3, A^h>0, be defined by: 

    A^h(x',x_3)=I_3+h^γS(x')+h^γ/2x_3B(x'),

where the scaling exponent γ>2 and I_n is the n× n identity matrix.  Consider a family of three-dimensional thin plates: 
    Ω^h=ω×(-h/2,h/2)⊂ℝ^3,
 where ω is an open bounded set of ℝ^2 and 0<h≪1, viewed as the reference configurations of thin elastic films.  A typical point in Ω^h is denoted by x=(x_1,x_2,x_3)=(x',x_3), where x'∈ω and |x_3|<h/2.  The “stretching" and “bending" tensors S,B:ω→ℝ^3×3 are two given smooth matrix fields.

	For a deformation u^h:Ω^h→ℝ^3, its elastic energy I_W^h(u^h), defined by:

    I_W^h(u^h)   =1/h∫_Ω^hW(F)dx
       =1/h∫_Ω^hW(∇ u^h(A^h)^-1)dx ∀ u^h∈ W^1,2(Ω^h,ℝ^3),
 
is given in terms of the elastic tensor F=∇ u^h(A^h)^-1 (see <cit.>) accounting for the reorganization of Ω^h in response to A^h.

	The elastic energy density W:ℝ^3× 3→ℝ_+ is assumed to satisfy the standard conditions of normalization, frame indifference with respect to the special orthogonal group SO(3) of proper rotations in ℝ^3, and second order nondegeneracy given by: 

    ∃  c>0  ∀  F∈ℝ^3× 3∀ R∈ SO(3)    W(R)=0,  W(RF)=W(F),
        W(F)≥ cdist^2(F,SO(3)).

	We also assume that there exists a monotone nonnegative function ν:[0,+∞]→[0,+∞] which converges to zero at 0, and a quadratic form 𝒬_3 on ℝ^3× 3, with:

    ∀ F∈ℝ^3× 3|W(I_3+F)-1/2𝒬_3(F)|≤ν(|F|)|F|^2.

	If W is 𝒞^2 regular in a neighborhood of SO(3), then condition (<ref>) is satisfied and, in that case, we have 𝒬_3=D^2W(I_3).  Note that (<ref>) implies that 𝒬_3 is nonnegative, is positive definite on symmetric matrices, and 𝒬_3(F)=𝒬_3(sym F) for all F∈ℝ^3× 3 (see <cit.> for a proof).

	Recall that in (<ref>), I_W^h(u^h)=0 is equivalent, via  (<ref>) and the polar decomposition theorem, to:

    (∇ u^h)^T∇ u^h=(A^h)^T(A^h) anddet∇ u^h>0 in Ω^h;

which is equivalent to: I_W^h(u^h)=0 if and only if u^h is an isometric immersion of the Riemannian metric G^h=(A^h)^T(A^h).  Therefore, the quantity:

    e_h=inf{I_W^h(u^h);u^h∈ W^1,2(Ω^h,ℝ^3)}

measures the residual energy at free equilibria of the configuration Ω^h that has been prestrained by G^h.  This is consistent with Theorem 2.2 in <cit.>, which observes that e_h>0 whenever G^h has no smooth isometric immersion in ℝ^3, i.e. when there is no u^h with (<ref>) or, equivalently, when the Riemann curvature tensor of the metric G^h does not vanish identically on Ω^h.

	Observe now that A^h in (<ref>) yields:

    G^h(x',x_3)   =(A^h)^T(A^h)
       =I_3+2h^γsym S(x')+2h^γ/2x_3symB(x')
         +2h^3γ/2x_3sym S(x')^TB(x')
         +h^2γS(x')^TS(x')+h^γx_3^2B(x')^TB(x').




 §.§ Notation.

For a matrix F, F_n× m denotes its n× m principal minor.  If m=n, the symmetric part of a square matrix F is denoted by symF=(F+F^T)/2.  The superscript ^T refers to the transpose of a matrix or an operator.  

	Also, for any F∈ℝ^2×2, we denote by F^*∈ℝ^3×3 the matrix for which F^*_2×2=F and F^*_i3=F^*_3i=0, for i=1,2,3.  By ∇_tan we denote taking derivatives ∂_1 and ∂_2 in the in-plate directions e_1=(1,0,0)^T and e_2=(0,1,0)^T.  The derivative ∂_3 is taken in the out-of-plate direction e_3=(0,0,1)^T.

	Finally, we use the Landau symbols 𝒪(h^α) and o(h^α) to denote quantities which are of the order of, or vanish faster than h^α, as h→0.  By C we denote any universal constant, depending on ω and W, but independent of other involved quantities, so that C=𝒪(1), and it can change values from line to line.



 §.§ Formal discussion about scaling.

	Consider the deformations u^h:Ω^h→ℝ^3 of the shell Ω^h given by: 
    u^h(x',x_3)=(x',0)^T+h^γ/2 V(x')+x_3N_h(x')+h.o.t.,
 where N_h(x') is the unit normal to the midplate and V:ω→ℝ^3.  Notice that we have:

    ∇ u^h(x',x_3)=(I_3)^*+h^γ/2∇ V+ N_h⊗ e_3+x_3∇ N_h+h.o.t.,

and, since F=∇ u^h (A^h)^-1, we have that:

    √(F^TF)   =I_3-h^γ sym  S-x_3h^γ/2 sym B+h^γ/2 sym∇ V
         +h^γ/2∇ V^T∇ V+x_3sym([I_3+h^γ/2∇ V^T]∇ N_h)+h.o.t.

Now, by (<ref>) and using the Taylor expansion for W, we can go back to (<ref>) to obtain:

    I_W^h(u^h)=1/h∫_Ω^hW(√(F^TF))dx'dx_3 =1/h∫_Ω^h(1/2𝒬_3(⋆)+h.o.t.)dx'dx_3,

where 𝒬_3(⋆)=D^2W(Id)(⋆,⋆), and ⋆ is given by:

    ⋆   =-h^γ sym S-x_3h^γ/2 symB+h^γ/2 sym∇ V
         +h^γ/2∇^TV∇ V+x_3sym([I_3+h^γ/2∇ V^T)]∇ N_h)+h.o.t.


Thus, we can rewrite (<ref>) as:

    I_W^h(u^h)   =∫_ω1/2𝒬_3(-h^γ symS+h^γ/2 sym∇ V+h^γ/2∇ V^T∇ V)dx'
         +h^2/24∫_ω𝒬_3(-h^γ/2 symB-h^γ/2∇^2V_3+h^γ𝒟)dx'+h.o.t.
       =h^2γ/8∫_ω𝒬_3(-2 symS+∇ V^T∇ V)dx'
         +h^γ+2/24∫_ω𝒬_3(- symB-∇^2V_3+h^γ/2𝒟)dx'+h.o.t.,
 
where the matrix D is given by:

    𝒟=[ V_3,1V_1,11+V_3,2V_2,11 V_3,1V_1,12+V_3,2V_2,12; V_3,1V_1,21+V_3,2V_2,21 V_3,1V_1,22+V_3,2V_2,22 ].

Since γ>2, we observe that I_W^h(u^h)≈ C h^γ+2 and hence, we expect the Γ-limit of 1/h^γ+2I_W^h(u^h) to be only the first order change in the linear bending energy: 
    1/24∫_ω𝒬_3(B+∇^2V_3) dx'.




§ THE VARIATIONAL LIMIT IN THE CASE Γ>2.


As explained before, the main goal of this paper is the identification of the asymptotic behavior of the minimizers of I_W^h(u^h) as h→0, through deriving the Γ-limit of the rescaled energies 1/h^γ+2I_W^h(u^h).

In <cit.> the authors derived a new variational model consisting of minimizing a biharmonic energy ∫_ω |∇^2v|^2 dx' of the out-of plane displacement v∈ W^2,2(ω,ℝ), satisfying the Monge-Ampère constraint ∇^2v=f, where f=- curl^T curl S_2×2 is the linearized Gauss curvature of the Riemannian metrics in (<ref>).  This work was done in the parameter range 0<γ<2, whereas the case γ=2 was treated in <cit.>, leading to the derivation of the Föppl-von Kármán equations accounting for presence of the prestrain.  In what follows, we carry out the analysis for the parameter range γ>2.  

We now state the main results of this paper:

	
	Let A^h be given as in (<ref>), with an arbitrary exponent γ>2.  Assume that a sequence of deformations u^h∈ W^1,2(Ω^h,ℝ^3) satisfies:

    I_W^h(u^h)≤ Ch^γ+2,

where W fulfills (<ref>) and (<ref>).  Then, there exist rotations R̅^h∈ SO(3) and translations c^h∈ℝ^3 such that, for the normalized deformations:

    y^h∈ W^1,2(Ω^1,ℝ^3),  y^h(x',x_3)=(R̅^h)^Tu^h(x',hx_3)-c^h,

the following hold (up to a subsequence that we do not relabel):

	
  (i) y^h(x',x_3)→ x' in W^1,2(Ω^1,ℝ^3)
	
  (ii) The scaled displacements 

    V^h(x')=1/h^γ/2_-1/2^1/2y^h(x',t)-x'dt
 
converge to a vector field V of the form V=(0,0,V_3)^T.  This convergence is strong in W^1,2(ω,ℝ^3).  The only non-zero out-of-plane scalar component V_3 of V belongs to W^2,2(ω,ℝ).  
	
  (iii) Moreover: 

    lim inf_h→01/h^γ+2I_W^h(u^h)≥ℐ_γ(V_3),

where ℐ_γ:W^2,2(ω)→ℝ̅_+ is given by:

    ℐ_γ=1/24∫_ω𝒬_2(∇^2 V_3+( symB(x'))_2×2) dx',

and the quadratic non-degenerate form 𝒬_2, acting on matrices F∈ℝ^2×2 is given by:

    𝒬_2(F)=min{𝒬_3(F̃):F̃∈ℝ^3×3, F̃_2×2=F }.




Now, for the optimality of the energy bound in (<ref>) and of the scaling (<ref>):

	  Assume (<ref>), (<ref>), and (<ref>).  Moreover, assume that ω is simply connected and that γ>2.  Then, for every V_3∈ W^2,2(ω,ℝ), there exists a sequence of deformations u^h∈ W^1,2(Ω^h,ℝ^3) such that the following hold:

	
  (i) The sequence y^h(x',x_3)=u^h(x',hx_3) converge in W^1,2(Ω^1,ℝ^3) to x'.
	
  (ii) V^h(x')=1/h^γ/2_-h/2^h/2(u^h(x',t)-x') dt converge in W^1,2(Ω,ℝ^3) to (0,0,V_3)^T.
	
  (iii) Recalling (<ref>), one has:

    lim_h→01/h^γ+2I_W^h(u^h)=ℐ_γ(V_3).




As a result of Theorem <ref> and Theorem <ref> we have:

	 
	Assume (<ref>), (<ref>), and (<ref>).  Moreover, assume that ω is simply connected and that γ>2.  Then there exist a uniform constant C≥0 such that: 
    e_h= inf I_W^h≤ Ch^γ+2.
  Under this condition, for any minimizing sequence u^h∈ W^1,2(Ω^h,ℝ^3) for I_W^h, i.e. when: 

    lim_h→01/h^γ+2(I_W^h(u^h)- infI_W^h)=0,

the convergences (i), (ii) of Theorem <ref> hold up to a subsequence, and the limit V_3 is a minimizer of the functional I_γ defined as in (<ref>).

	Moreover, for any (global) minimizer V_3 of I_γ, there exists a minimizing sequence u^h, satisfying (<ref>) together with (i), (ii), and (iii) of Theorem <ref>.




§ PROOF OF THEOREM <REF>.



This proof will be separated in different sections.  In Section <ref>, we obtain the rotations R̅^h required by Theorem <ref>.  In Sections <ref>, <ref>, and <ref> we present the proof of statements (i), (ii), and (iii) respectively.


 §.§ Construction of the rotations R̅^h.
 

First, we quote the following approximation result (Theorem <ref>), which can be directly obtained from the geometric rigidity estimate found in Theorem 1.6 in <cit.>, in view of the following bounds:

    .Var(A^h)=∇_t(A^h|_x_3=0)_L^∞(ω)+∂_3A^h_L^∞(Ω^h)≤ Ch^γ/2,
       A^h_L^∞(Ω^h)+(A^h)^-1_L^∞(Ω^h)≤ C.



	Let u^h∈ W^1,2(Ω^h,ℝ^3) satisfy lim_h→01/h^2I_W^h(u^h)=0, which is, in particular, implied by (<ref>).  Then, there exist matrix fields R^h∈ W^1,2(ω,ℝ^3×3), such that R^h(x')∈ SO(3) for a.e. x'∈ω, and:

    1/h∫_Ω^h|∇ u^h(x)-R^h(x')A^h(x)|^2dx≤ Ch^2+γ, ∫_ω|∇ R^h|^2 dx'≤ Ch^γ.


  In order to achieve the proof of compactness in Theorem <ref>, we outline similar arguments as in <cit.>, emphasizing the differences and new outcomes of this paper.

	Assume (<ref>) and let R^h∈ W^1,2(ω,ℝ^3×3) be the matrix fields given by Theorem <ref>. Define the averaged rotations: R̃^h=ℙ_SO(3)_ωR^h.  These projections of _ωR^h onto SO(3) are well defined for small h since, by (<ref>) and Poincaré's inequality, we have:

    dist^2(_ωR^h,SO(3))≤ C_ω|∇ R^h|^2 dx'≤ Ch^γ,

which together with Poincaré's inequality, deliver:

    ∫_ω|R^h-R̃^h|^2≤ C(∫_ω|R^h-_ωR^h|^2+ dist^2(_ωR^h,SO(3)))≤ Ch^γ.

	Now, let:

    R̂^h=ℙ_SO(3)_Ω^h(R̃^h)^T∇ u^h,

which is well defined for small h because:

    dist^2   (_Ω^h(R̃^h)^T∇ u^h,SO(3))
       ≤ C(_Ω^h|∇ u^h-R^hA^h|^2+_Ω^h|A^h-I_3|^2+_Ω^h|R^h-R̃^h|^2)
       ≤ Ch^γ,

where we used (<ref>), (<ref>), and (<ref>).  Consequently, using the fact that I_3=ℙ_SO(3)I_3, we also obtain: 

    |R̂^h-I_3|^2   ≤ C( dist^2(_Ω^h(R̃^h)^T∇ u^h,SO(3))+|_Ω^h(R̃^h)^T∇ u^h-I_3|^2) 
       ≤ Ch^γ.

We may now define:

    R̅^h=R̃^hR̂^h.


By (<ref>), (<ref>), and (<ref>), it follows that:

    ∫_ω|R^h-R̅^h|^2≤ C(∫_ω|R^h-R̃^h|^2+∫_ω|R̃^h(I_3-R̂^h)|^2)≤ Ch^γ.


Now, consider:

    (R̅^h)^TR^h-I_3^2_W^1,2(ω)=∫_ω|R^h-R̅^h|^2+∫_ω|∇ R^h|^2≤ Ch^γ,

and observe that from (<ref>) we conclude:

    lim_h→0(R̅^h)^TR^h=I_3  in W^1,2(ω,ℝ^3×3).



There exist vectors c^h∈ℝ^3 such that, for the re-scaled averaged displacement V^h defined as in (<ref>): 
    V^h(x')=1/h^γ/2_-1/2^1/2(R̅^h)^Tu^h(x',ht)-c^h-x'dt,
 it holds:
	
    ∫_ωV^h dx'=0, skew∫_ω∇ V^h dx'=0.

	

To ensure the first statement in (<ref>), we can take: 
    c^h=1/|ω|∫_ω_-1/2^1/2[(R̅^h)^Tu^h(x',ht)-x']dtdx'.

For the second statement in (<ref>), we observe that: 

    ∇ V^h=1/h^γ/2_-1/2^1/2[(R̅^h)^T∇_tanu^h(x',ht)-(I_3)^*]dt.

In view of (<ref>), we obtain that: 
    (R̅^h)^T_Ω_h∇ u^h=(R̂^h)^T_Ω^h(R̃^h)^T∇ u^h;
 and, because of (<ref>) and (<ref>), (R̅^h)^T_Ω_h∇ u^h is symmetric.  Hence: 
    skew_ω∇ V^h=h^-γ/2skew_Ω^h(R̅^h)^T∇ u^h=0.
  In particular, we see as well that (<ref>) is equivalent to:

    R̅^h=ℙ_SO(3)_Ω^h∇ u^h.

The above result is true since for a matrix F sufficiently close to SO(3), its projection R_0=ℙ_SO(3)F coincides with the unique rotation appearing in the polar decomposition of F, that is, F=R_0U with skew U=0. 



 §.§ Proof of (i) in Theorem <ref>.

  We use (<ref>), (<ref>), (<ref>), and (<ref>), to obtain:

    (∇ y^h-I_3)_3×2^2_L^2(Ω^1)
         ≤1/h∫_Ω^h|(R̅^h)^T∇ u^h-I_3|^2
         ≤ C(1/h∫_Ω^h|(R̃^h)^T∇ u^h-I_3|^2+1/h∫_Ω^h|R̂^h-I_3|^2)
         ≤ Ch^γ,

and also, by (<ref>), we get:

    ∂_3 y^h^2_L^2(Ω^1)   =h∫_Ω^h|∂_3 u^h|^2
       ≤ Ch(∫_Ω^h|∇ u^h-R^hA^h|^2+∫_Ω^h|A^h|^2)
       ≤ Ch^2.

By the first statement in (<ref>), we have:

    ∫_Ω^1y^h(x)-x' dx=h^γ/2∫_ωV^h dx'=0;

hence, using Poincaré's inequality, (<ref>), and (<ref>), we have:

    y^h(x)-x'^2_L^2(Ω^1)   ≤ C∫_Ω^1|∇ y^h-I_3|^2
       ≤ C(∫_Ω^1|∇ y^h-I_3|_3×2^2+∫_Ω^1|∂_3 y^h|^2)
       ≤ Ch^2,

and therefore, we obtain the convergence of y^h to x' in W^1,2(Ω^1).



 §.§ Proof of (ii) in Theorem <ref>.



V^h converges (up to a subsequence) weakly in W^1,2(ω,ℝ^3).
 
Observe that, using the first statement in (<ref>) and Poincaré's inequality, we have:

    V^h^2_L^2(ω)≤ C∇ V^h^2_L^2(ω).

Also, by Jensen's inequality and (<ref>), we obtain:

    ∇ V^h^2_L^2(ω)   =||1/h^γ/2_-1/2^1/2[(R̅^h)^T∇_tanu^h(x',ht)-(I_3)^*]dt||^2_L^2(ω)
       ≤C/h^γ∫_Ω^1|(R̅^h)^T∇_tanu^h(x',ht)-(I_3)^*|^2
       ≤ C.

Therefore:

    V^h^2_W^1,2(ω)=V^h^2_L^2(ω)+∇ V^h^2_L^2(ω)≤ C.

Hence, V^h is a bounded sequence in the norm of W^1,2(ω,ℝ^3), which implies the V^h is weakly convergent (up to a subsequence, still called V^h) to V in W^1,2(ω,ℝ^3).  

	Consider the matrix fields D^h∈ W^1,2(ω,ℝ^3×3):

    D^h(x')   =1/h^γ/2_-h/2^h/2(R̅^h)^TR^h(x')A^h(x',t)-I_3  dt
       = h^γ/2(R̅^h)^TR^h(x') S(x')+1/h^γ/2((R̅^h)^TR^h(x')-I_3).



The sequence D^h converges (up to a subsequence) weakly in W^1,2(ω,ℝ^3×3).
 

	Since S is smooth and by (<ref>), we have that:

    D^h_L^2(ω)^2
         =∫_ω|h^γ/2(R̅^h)^TR^h(x') S(x')+1/h^γ/2((R̅^h)^TR^h(x')-I_3)|^2 dx'
         ≤ C(h^γ∫_ω|S(x')|^2 dx'+1/h^γ∫_ω|(R^h(x')-R̅^h)|^2 dx')
         ≤ C.

Also, by (<ref>), we have that:

    ∇ D^h_L^2(ω)^2
       =∫_ω|h^γ/2(R̅^h)^T∇(R^h(x') S(x'))+1/h^γ/2(R̅^h)^T∇ R^h(x')|^2 dx'
       ≤ C(h^γ∫_ω|∇ R^h(x')|^2| S(x')|^2 dx'
         +h^γ∫_ω|∇ S(x')|^2 dx'+1/h^γ∫_Ω|∇ R^h|^2)
       ≤ C.

Therefore D^h^2_W^1,2(ω)≤ C and so, up to a subsequence still called D^h, we obtain:

    lim_h→0D^h=D weakly in W^1,2(ω,ℝ^3×3)


The following limit:

    lim_h→01/h^γ/2((R̅^h)^TR^h-I_3)=D in L^q(ω,ℝ^3×3),

for all q≥1, is obtained by noticing that:

    ||1/h^γ/2((R̅^h)^TR^h-I_3)-D||_L^q(ω)
         ≤||1/h^γ/2((R̅^h)^TR^h-I_3)-D^h||_L^q(ω)+D^h-D_L^q(ω)
         ≤ h^γ/2(R̅^h)^TR^h(x') S(x')_L^q(ω)+D^h-D_L^q(ω).

Since SO(3) is a bounded set, we know that (R̅^h)^TR^h_L^∞(ω)≤ C for some C>0; and since S is smooth, we have that S_L^q(ω) is also bounded.  Then, the first term on the right hand side of (<ref>) is bounded by Ch^γ/2.  For the second term, by Lemma <ref>, the Rellich-Kondrachov theorem, and since ω is bounded, we have that D^h converges strongly in L^q(Ω) to D, for q≥1 .  Therefore we have proven (<ref>).


The limiting matrix field D has skew-symmetric values:

    sym D=lim_h→0 sym D^h=0.



For all R∈ SO(3), we have:

    (R-I_3)^T(R-I_3)=2 I_3-2  sym R=-2  sym(R-I_3).

Now, using (<ref>), (<ref>), and (<ref>), we obtain:

    1/h^γ/2 sym((R̅^h)^TR^h-I_3)_L^2(ω)
         =1/2h^γ/2((R̅^h)^TR^h-Id)^T((R̅^h)^TR^h-I_3)_L^2(ω)
         ≤ C1/h^γ/2R^h-R̅^h^2_W^1,2(ω)
         ≤ Ch^γ/2.
	
Then, it follows that: 
    1/h^γ/2 sym((R̅^h)^TR^h-I_3)_L^2(ω)→0
 as h→0; and, as a result, D has skew-symmetric values.  


By (<ref>), we observe that:

    1/h^γ/2 symD^h   = sym[(R̅^h)^TR^h(x') S(x')+1/h^γ((R̅^h)^TR^h(x')-I_3)]
       = sym[(R̅^h)^TR^h(x') S(x')]
         -1/2h^γ((R̅^h)^TR^h-I_3)^T((R̅^h)^TR^h-I_3).


First, since S is smooth, we have:

    (R̅^h)^TR^h(x') S(x')-S(x')_L^q(ω)   =[(R̅^h)^TR^h(x')-I_3]S(x')_L^q(ω)
       ≤ C(R̅^h)^TR^h(x')-I_3_L^q(ω).

Using (<ref>) and the Rellich-Kondrachov theorem, it follows that the limit, as h→0, of the right hand side of (<ref>) is zero, for q≥1.

Also, observe that:

    ||1/h^γ((R̅^h)^TR^h-I_3)^T((R̅^h)^TR^h-I_3)+D^2||_L^q(ω)
       ≤||1/h^γ/2((R̅^h)^TR^h-I_3)^T[1/h^γ/2((R̅^h)^TR^h-I_3)-D]||_L^q(ω)
         +||[1/h^γ/2((R̅^h)^TR^h-I_3)-D]^TD||_L^q(ω)

approaches 0, as h→0, by (<ref>).  

Therefore, by (<ref>), (<ref>) and (<ref>), we obtain:

    lim_h→01/h^γ/2 symD^h   =lim_h→0 sym[(R̅^h)^TR^h(x') S(x')]
         -1/2h^γ((R̅^h)^TR^h-I_3)^T((R̅^h)^TR^h-I_3)
       = sym S+1/2D^2  in L^q(ω,ℝ^3×3),   ∀ q≥1.


As for the convergence of V^h, we have by (<ref>) and (<ref>) that:

    [∇ V^h(x')-D^h(x')]_3×2
       =1/h^γ/2(R̅^h)^T_-h/2^h/2[∇_tanu^h(x',t)-R^h(x')A^h(x',t)]_3×2dt,

which, together with (<ref>), imply that:

    [∇ V^h(x')-D^h(x')]_3×2^2_L^2(ω)
       ≤C/h^γ+1∫_Ω^h|∇ u^h(x',t)-R^h(x')A^h(x',t)|^2dx
       ≤ Ch^2,

and therefore, by (<ref>), the sequence ∇ V^h converges in L^2(ω,ℝ^3×2) to D.  
Observe that:

    ∇_tan V=D_3×2,

and D∈ W^1,2(ω,ℝ^3×3) by (<ref>).  Then, we obtain V∈ W^2,2(ω,ℝ^3).

Finally, we use (<ref>) to conclude that sym(∇_tan V)_2×2=0, and so, by Korn's inequality:

    ∫_ω|(∇_tan V)_2×2|^2dx'≤ C∫_ω| sym(∇_tan V)_2×2|^2dx'=0,

which implies V_tan must be constant and, hence, equal to 0 in view of (<ref>).  Then V=(0,0,V_3).  This ends the proof of (ii) in Theorem <ref>.



 §.§ Proof of (iii) in Theorem <ref>.

 To prove the lower bound (<ref>), we define the rescaled strains P^h∈ L^2(Ω^1,ℝ^3×3) by:

    P^h(x',x_3)=1/h^γ/2+1((R^h(x'))^T∇ u^h(x',hx_3)(A^h(x',hx_3))^-1-I_3).



The rescaled strains P^h converge (up to a subsequence) weakly in L^2(Ω^1,ℝ^3×3).
 

	Observe that, since A^h is smooth, R^h∈ SO(3), and using (<ref>), we have:

    P^h^2_L^2(Ω^1)≤C/h^γ+3∫_Ω^h|∇ u^h(x',t)-A^h(x',t)|^2dx'dt≤ C.

Then, up to a subsequence, we get:

    lim_h→0P^h=Pweakly in L^2(Ω^1,ℝ^3×3).


	We follow similar steps as in <cit.> to obtain a useful property of the limiting strain P.  First, by (<ref>), we obtain:

    (∂_3y^h-he_3)/h^γ/2+1   =1/h^γ/2(R̅^h)^T[∇ u^h(x',hx_3)-R^h(x')A^h(x',hx_3)]e_3
         +1/h^γ/2(R̅^h)^TR^h(x')[A^h(x',hx_3)-I_3]e_3
         +1/h^γ/2[(R̅^h)^TR^h(x')-I_3]e_3.


Let's study all three terms on the right-hand side of (<ref>).  For the first term, using (<ref>), we get:

    1/h^γ   ∫_Ω^1|(R̅^h)^T[∇ u^h(x',hx_3)-R^h(x')A^h(x',hx_3)]e_3|^2dx'dx_3
       ≤1/h^γ+1∫_Ω^h|∇ u^h(x',t)-R^h(x')A^h(x',t)|^2dx'dx_3
       ≤ Ch^2.

For the second term, we have:

    1/h^γ   ∫_Ω^1|(R̅^h)^TR^h(x')[A^h(x',hx_3)-I_3]e_3|^2dx'dx_3
       ≤∫_Ω^1|h^γ/2 S(x')+hx_3B(x')|^2dx'dx_3
       ≤ Ch^2.

And finally, for the third term, using (<ref>), we have that it converges to De_3 in L^2(ω).  Therefore:

    lim_h→01/h^γ/2+1(∂_3y^h-he_3)=De_3  in L^2(Ω^1, ℝ^3).


	For each small s>0, we define the sequence of functions f^s,h∈ W^1,2(Ω^1,ℝ^3):

    f^s,h(x)=1/h^γ/2+11/s(y^h(x+se_3)-y^h(x)-hse_3).

Clearly, (<ref>) is equivalent to 
    f^s,h(x)=1/h^γ/2+1_0^s(∂_3y^h(x+te_3)-he_3)dt,
 and, by (<ref>), we have that:

    lim_h→0f^s,h=De_3  in L^2(Ω^1,ℝ^3).

Also, observe that ∂_3f^s,h(x)=1/s1/h^γ/2+1(∂_3y^h(x+se_3)-∂_3y^h(x)).  Then:

    ∂_3f^s,h(x)_L^2(Ω^1)   ≤1/|s|1/h^γ/2+1(∂_3y^h(x+se_3)-he_3)-De_3_L^2(Ω^1)
         +1/|s|1/h^γ/2+1(∂_3y^h(x)-he_3)-De_3_L^2(Ω^1)

goes to 0 as h→0, by (<ref>).    In other words:

    lim_h→0∂_3f^s,h=0  in L^2(Ω^1,ℝ^3).

Further, for any α=1,2, we have:

    ∂_α f^s,h(x)   =1/s(R̅^h)^TR^h(x')[P^h(x',x_3+s)A^h(x',hx_3+hs)
         -P^h(x',x_3)A^h(x',hx_3)
         +1/h^γ/2+1(A^h(x',hx_3+hs)-A^h(x',hx_3))]e_α;

which, in view of (<ref>), (<ref>), and since A^h→ I_3 strongly in L^2(ω,ℝ^3×3), yields the weak convergence in L^2(Ω^1,ℝ^3×2) of ∂_α f^s,h(x):

    lim_h→0∂_α f^s,h(x)   =1/s(P(x',x_3+s)-P(x',x_3))e_α
         +1/slim_h→01/h^γ/2+1(I_3+h^γ S(x')+h^γ/2+1(x_3+s)B(x').
         .-(I_3+h^γ S(x')+h^γ/2+1x_3B(x')))e_α
       =1/s(P(x',x_3+s)-P(x',x_3))e_α+B(x')e_α.

Consequently, by (<ref>), (<ref>), and (<ref>), we see that f^s,h converges weakly in W^1,2(ω,ℝ^3) to De_3.  Hence, the left-hand side in (<ref>) equals ∂_α(De_3):

    ∂_α(De_3)=1/s(P(x',x_3+s)-P(x',x_3))e_α+B(x')e_α.

Also, we notice that:

    (∂_3 P)e_α=lim_s→0(P(x',x_3+s)-P(x',x_3)/s)e_α=∂_α(De_3)-B(x')e_α;

and consequently:

    P(x)_3×2=(∇(D(x')e_3)-B(x'))_3×2x_3+P_0(x')_3×2,

for some P_0∈ L^2(ω,ℝ^3×3).

We can now finish the proof of Theorem <ref>.   Observe that, from (<ref>), and using the Taylor expansion of the function W(F) close to F=I_3, we obtain:

    1/h^γ+2   W(∇ u^h(x)(A^h(x))^-1)
       =1/h^γ+2W((R^h(x))^T∇ u^h(x)(A^h(x))^-1)
       =1/h^γ+2W(Id+h^γ/2+1P^h(x))
       =1/2𝒬_3(P^h(x))+ν(h^γ/2+1|P^h|)𝒪(|P^h(x)|^2).

Consider the sets 𝒰_h={x∈Ω^1 : h|P^h(x',x_3)|≤1}.  Note that, using Hölder's inequality and (<ref>), we have:

    ∫_Ω^1|hP^h|dx
         =1/h^γ/2+1∫_Ω^h|(R^h(x'))^T∇ u^h(x',t)(A^h(x',t))^-1-Id|dx
         ≤C/h^γ/2+1(∫_Ω^h|∇ u^h(x',hx_3)-R^h(x')A^h(x',hx_3)|^2dx)^1/2
         ≤ Ch^1/2.

Since, by (<ref>), hP^h converges to 0 in L^1(Ω^1) as h→0, then there exists a subsequence (which we call again hP^h) such that hP^h converges to 0 pointwise a.e.

Now, for that subsequence observe that, by (<ref>), we obtain:

    ∫_Ω^1|χ_𝒰_h-1|^2dx=∫_Ω^1∖𝒰_h1dx
         ≤∫_Ω^1∖𝒰_h|hP^h|dx≤∫_Ω^1|hP^h|dx≤ Ch^1/2.

And therefore, χ_𝒰_h converges to 1 in L^2(Ω^1) as h→0.

	Since lim_t→0ν(t)=0, using (<ref>), and the properties of 𝒬_3, we get:

    lim inf_h→01/h^γ+2I_W^h(u^h)
         ≥lim inf_h→01/h^γ+2∫_Ω^1χ_𝒰_hW(∇ u^h(x',hx_3)(A^h(x',hx_3))^-1)dx
         =lim inf_h→0(1/2∫_Ω^1𝒬_3(χ_𝒰_hP^h(x))dx+o(1)∫_Ω^1|P^h(x)|^2dx)
         ≥1/2∫_Ω^1𝒬_3( symP(x))dx,

where the convergence to 0 of the term o(1)∫_Ω^1|P^h(x)|^2dx as h→0 follows from (<ref>);  and since, by (<ref>), χ_𝒰_hP^h(x) converges weakly to P in L^2(Ω^1,ℝ^3×3).  

	Further, by (<ref>) and (<ref>):

    1/2   ∫_Ω^1𝒬_3( symP(x))dx
       ≥1/2∫_Ω^1𝒬_2( symP_2×2(x))dx
       =1/2∫_Ω^1𝒬_2(x_3  sym(∇ De_3-B(x'))_2×2+ symP_0(x')_2×2)dx
       ≥1/24∫_Ω𝒬_2( sym(∇ De_3)_2×2-( symB(x'))_2×2) dx'.

Now, in view of Theorem <ref> (ii), (<ref>), and (<ref>) we have: 
    (De_3)^T=⟨ D_1,3, D_2,3, 0⟩^T=⟨ -D_3,1, -D_3,2, 0⟩^T=-⟨∂_1V_3, ∂_2V_3, 0⟩^T,
 in other words: 
    (∇ De_3)_2×2=-∇^2V_3,
 which yields the claim in Theorem <ref> (iii), by (<ref>) and (<ref>).



§ PROOF OF THEOREM <REF>.


Recalling (<ref>) the definition of 𝒬_2, let c(F)∈ℝ^3 be the unique vector so that: 
    𝒬_2(F)=𝒬_3(F^*+ sym(c⊗ e_3)).
 
The mapping c:ℝ^2×2_ sym→ℝ^3 is well-defined and linear, by the properties of 𝒬_3.   Also, for all F∈ℝ^3×3, we denote by l(F) the unique vector in ℝ^3, linearly depending on F, such that: 
    sym(F-(F_2×2)^*)= sym(l(F)⊗ e_3).


	Let the out-of-plane displacement V_3 be as in Theorem <ref>.  By the Rellich-Kondrachov embedding theorem, since V_3∈ W^2,2(ω,ℝ), we have V_3∈ W^1,q(ω,ℝ) for all 1≤ q<∞.  We first prove the result under the additional assumption of V_3 being smooth up to the boundary in Section <ref>.  In Section <ref>, we prove the result for V_3∈ W^2,2(ω,ℝ).



 §.§ Case: V_3∈ C^∞(ω̅,ℝ).

  Define the recovery sequence:

    u^h(x',x_3)
       =[ x';  0 ]+[            0; h^γ/2V_3(x') ]+x_3[ -h^γ/2 (∇_tan V_3(x'))^T;                        1 ]+1/2h^γ/2x_3^2d^1(x'),

for (x',x_3)∈Ω^h, where the smooth warping field d^1:Ω̅→ℝ^3 is given by:

    d^1=l(B)+c(-∇^2 V_3-( sym B)_2×2).

Calculating the deformation gradient, we obtain:

    ∇ u^h=I_3+h^γ/2𝒱-h^γ/2x_3(∇_tan^2V_3)^*+h^γ/2[ 1/2x_3^2∇_tan d^1            x_3d^1 ],

where the skew-symmetric matrix field 𝒱 is given by

    𝒱=[              0 -(∇_tan V_3)^T;      ∇_tan V_3              0 ].

The convergence statements in (i) and (ii) of Theorem <ref> are verified by a straightforward calculation:

	For (i):

    y^h(x',x_3)-x'_W^1,2(Ω^1,ℝ^3)≤ Ch.


	For (ii):

    V^h(x')-[   0   0 V_3 ]^T_W^1,2(Ω,ℝ^3)≤ Ch^2.


 To prove (iii), we need to estimate the energy of the sequence u^h.   In order to do this, we shall use an auxiliary SO(3)-valued matrix R^h=e^h^γ/2𝒱.  Clearly, R^h=I_3+h^γ/2𝒱+h^γ/2𝒱^2+h.o.t and (R^h)^T=I_3-h^γ/2𝒱+h^γ/2𝒱^2+h.o.t.  Also, recall that (A^h)^-1=Id-h^γS-h^γ/2x_3B+h.o.t.  Hence, we obtain: 

    (R^h)^T(∇ u^h)(A^h)^-1
       =I_3+h^γ(-1/2𝒱^2-S)+h^γ/2x_3(-(∇^2V_3)^*-B+d^1⊗ e_3)+h.o.t.


	Using the definition of the quadratic form 𝒬_3(F), Taylor expanding the energy density W around the identity, and taking into account the uniform boundedness of all the involved functions and their derivatives, we get in (<ref>):

    I_W^h(u^h)   =1/h∫_Ω^hW((R^h)^T(∇ u^h)(A^h)^-1)dx
       =h^2γ/2∫_Ω𝒬_3( sym(-1/2𝒱^2-S))dx'
         +h^γ+2/24∫_Ω𝒬_3( sym(-(∇^2V_3)^*-B+d^1⊗ e_3))dx'+h.o.t.

In view of (<ref>), it follows that:

    1/h^γ+2I_W^h(u^h)
         =1/24∫_Ω𝒬_3( sym(-(∇^2V_3)^*-B+d^1⊗ e_3))dx'+O(h^γ-2).


	Observe that, from (<ref>),we get that:

    sym(-(∇^2V_3)^*-B+d^1⊗ e_3)
       =(-∇^2V_3-( symB)_2×2)^*+ sym((d^1-l(B))⊗ e_3)
       =(-∇^2V_3-( symB)_2×2)^*+ sym((c(-∇^2 V_3-( sym B)_2×2))⊗ e_3),

which implies:

    1/h^γ+2I_W^h(u^h)=ℐ_f(V_3)+O(h^γ-2),

which, in turn, proves (iii) for smooth displacement V_3.



 §.§ Case: V_3∈ W^2,2(ω,ℝ).


We now consider a sequence V_3^n∈ C^∞(ω,ℝ) such that:

    V_3^n-V_3_W^2,2(ω,ℝ)→0,  as n→∞.

Define the sequence u^h,n as in (<ref>) using V_3^n instead of V_3:

    u^h,n(x',x_3)
       =[ x';  0 ]+[              0; h^γ/2V_3^n(x') ]+x_3[ -h^γ/2 (∇ V_3^n(x'))^T;                      1 ]+1/2h^γ/2x_3^2d^1,n(x'),

where the smooth warping field d^1,n:ω̅→ℝ^3 is given by:

    d^1,n=l(B)+c(-∇^2 V_3^n-( sym B)_2×2).

Calculating the deformation gradient, we first obtain:

    ∇ u^h,n= I_3+h^γ/2𝒱^n-h^γ/2x_3(∇^2V_3^n)^*+h^γ/2[ 1/2x_3^2∇ d^1,n        x_3d^1,n ],
 
where the skew-symmetric matrix field 𝒱^h is given by:

    𝒱^n=[            0 -(∇ V_3^n)^T;      ∇ V_3^n            0 ].


	Let the sequence {n(h)} be such that n(h)→∞ when h→0, and define:

    u_h(x',x_3):=u^h,n(h)(x',x_3).

Observe that:

    |1/h^γ+2I_W^h(u_h)-ℐ_f(V_3)|
         ≤|1/h^γ+2I_W^h(u_h)-ℐ_f(V_3^n(h))|+|ℐ_f(V_3^n(h))-ℐ_f(V_3)|.

We will study the two terms on the right hand side of (<ref>).  As above, we use an auxiliary SO(3)-valued matrix R^h,n(h)=e^h^γ/2𝒱^n(h).  Hence, we obtain: 

    I_3-(R^h,n(h))^T(∇ u_h)(A^h)^-1_L^∞(Ω^h,ℝ^3×3)
       ≤ C h^γ/2+1(1+||∇V_3^n(h)||_L^∞(ω)+||(∇^2V_3^n(h))^*||_L^∞(ω)
         +||∇^3V_3^n(h)||_L^∞(ω)).

Let ϵ>0, we can always find h_1>0 such that for all h≤ h_1: 

    h^γ/2+1(1+||∇V_3^n(h)||_L^∞(ω)+||(∇^2V_3^n(h))^*||_L^∞(ω)
         +||∇^3V_3^n(h)||_L^∞(ω))<ϵ/2C,

by reparameterizing the sequence V_3^n(h), in order to slow down the rate of convergence to V_3.  

Then, in a similar way to (<ref>), we obtain:

    1/h^γ+2I_W^h(u_h)
       =ℐ_f(V_3^n(n))+O(min(h^γ-2,h^2)F(∇V_3^n(h),(∇^2V_3^n(h))^*,∇^3V_3^n(h))),

where the quantity F(∇V_3^n(h),∇^2V_3^n(h),∇^3V_3^n(h)) depends only on L^∞(ω)-norms of ∇V_3^n(h), (∇^2V_3^n(h))^* and ∇^3V_3^n(h).  

As above, we can always find h_2>0 such that for all h≤ h_2:

    O(min(h^γ-2,h^2)|F(∇V_3^n(h),(∇^2V_3^n(h))^*,∇^3V_3^n(h))|)<ϵ/2,

which is possible by slowing down the rate of convergence of the sequence V_3^n(n) (by reparametrizing the sequence).  Take h_3=min(h_1,h_2).  Then, for the first term on the right hand side of (<ref>), we have: 

    |1/h^γ+2I_W^h(u_h)-ℐ_f(V_3^n(h))|<ϵ/2,

for all h≤ h_3.

Also, by (<ref>) and (<ref>), we obtain for the second term on the right hand side of (<ref>):

    |ℐ_f(V_3^n(h))-ℐ_f(V_3)|   ≤1/24∫_ω|𝒬_2(∇^2 V_3^n(h)+( symB(x'))_2×2)
         -𝒬_2(∇^2 V_3+( symB(x'))_2×2)| dx'
       =1/24∫_ω|𝒬_3((-∇^2V_3^n(h)-( symB)_2×2)^*
         + sym((c(-∇^2 V_3^n(h)-( sym B)_2×2))⊗ e_3))
         -𝒬_3((-∇^2V_3-( symB)_2×2)^*
         + sym((c(-∇^2 V_3-( sym B)_2×2))⊗ e_3))| dx'.

	Define: 

    T^n(h)   =(-∇^2V_3^n(h)-( symB)_2×2)^*
         + sym((c(-∇^2 V_3^n(h)-( sym B)_2×2))⊗ e_3),
 and: 

    T   =(-∇^2V_3-( symB)_2×2)^*
         + sym((c(-∇^2 V_3-( sym B)_2×2))⊗ e_3),
 
and notice that: 
    T^n(h)_L^2(ω)+T_L^2(ω)≤ C.

Then, using Hölder's inequality and properties of 𝒬_3, we obtain:

    |ℐ_f(V_3^n(h))-ℐ_f(V_3)|
       ≤1/24∫_ω|𝒬_3(T^n(h))-𝒬_3(T)| dx'
    
    
    	   ≤ C||T^n(h)-T||_L^2(ω)(||T^n(h)||_L^2(ω)+||T||_L^2(ω))
       ≤ C||T^n(h)-T||_L^2(ω),

Now, by (<ref>), there exists h_4>0 such that

    T^n(h)-T_L^2(ω)   =||(∇^2V_3^n(h)-∇^2V_3)^*
         + sym((c(∇^2 V_3^n(h)-∇^2 V_3))⊗ e_3)||_L^2(ω)
       ≤||(∇^2V_3^n(h)-∇^2V_3)^*||_L^2(ω)
         +|| sym((c(∇^2 V_3^n(h)-∇^2 V_3))⊗ e_3)||_L^2(ω)
       <ϵ/2C,

for all h≤ h_4.  Finally, taking h^*=min(h_3,h_4), we obtain that for all h≤ h^* we have: 
    |1/h^γ+2I_W^h(u_h)-ℐ_f(V_3)|<ϵ.
  This concludes the proof of (iii) in Theorem <ref>.



§ ACKNOWLEDGMENT.

The authors would like to thank Marta Lewicka for bringing this problem to their attention and for useful discussions and feedback.

plain



