Linear elasticity obtained from finite elasticity by Γ-convergence under weak coerciveness conditions

Ann. I. H. Poincaré – AN 29 (2012) 715–735

www.elsevier.com/locate/anihpc

Linear elasticity obtained from finite elasticity by Γ -convergence under weak coerciveness conditions

Virginia Agostiniani

, Gianni Dal Maso, Antonio DeSimone

SISSA, via Bonomea 265, 34136 Trieste, Italy

Received 23 June 2011; received in revised form 15 March 2012; accepted 17 April 2012 Available online 23 April 2012

Abstract

The energy functional of linear elasticity is obtained asΓ-limit of suitable rescalings of the energies of finite elasticity. The quadratic control from below of the energy densityW (∇v)for large values of the deformation gradient∇vis replaced here by the weaker conditionW (∇v)|∇v|^p, for somep >1. Energies of this type are commonly used in the study of a large class of compressible rubber-like materials.

©2012 Elsevier Masson SAS. All rights reserved.

1. Introduction

Consider an elastic body occupying a reference configurationΩ⊆Rⁿ, withn2, subject to some deformation v:Ω→Rⁿ. Assuming that the body is homogeneous and hyperelastic, the stored energy can be written as

Ω

W (∇v) dx,

where∇vis the deformation gradient and the energy densityW (F )0 is defined for everyF ∈R^n×nand is finite only for detF >0. We assume that the energy densityWis minimized at the value 0 by the identity matrixI, which amounts to saying that the reference configuration is stress free. We assume also thatW is frame indifferent, i.e., W (F )=W (RF )for everyF ∈R^n×nand everyRin the spaceSO(n)of rotations.

Since the deformationv(x)=xis an equilibrium when no external loads are applied, we expect that small external loadsεl(x)will produce deformations of the formv(x)=x+εu(x), so that the total energy is given by

Ω

W (I+ε∇u) dx−ε²

Ω

lu dx. (1.1)

In the case∇ubounded, by Taylor-expandingW (I+ε∇u)aroundI and rescaling (1.1) byε⁻², we obtain in the limitε→0 the formula

* Corresponding author.

E-mail addresses:vagostin@sissa.it(V. Agostiniani),dalmaso@sissa.it(G. Dal Maso),desimone@sissa.it(A. DeSimone).

0294-1449/$ – see front matter ©2012 Elsevier Masson SAS. All rights reserved.

http://dx.doi.org/10.1016/j.anihpc.2012.04.001

1 2

Ω

D²W (I )[∇u]²dx−

Ω

lu dx, (1.2)

whereD²W (I )[F]² is the second differential ofW atI applied to the pair[F, F]. By frame indifference, the first summand in (1.2) depends only on the symmetric parte(u)of the displacement gradient∇u, i.e.,

1 2

Ω

D²W (I )[∇u]²dx=1 2

Ω

D²W (I ) e(u)2

dx.

This functional is the linearized elastic energy associated with the displacementu.

This elementary derivation of linear elasticity requires onlyC²regularity ofW nearI, and hence in a neighbour- hood ofSO(n), by frame indifference. However, it does not guarantee that the minimizers of the most natural boundary value problems for (1.1) converge to the minimizer of the corresponding problems for the limit functional (1.2).

Convergence of minimizers has been established in[5]in the framework ofΓ-convergence, under the assumption W (F )d

F,SO(n)2

, (1.3)

whered(F,SO(n))is the distance ofF fromSO(n). The main result of the present paper is that the same conclusion holds if (1.3) is satisfied only in a neighbourhood ofSO(n), while the weaker condition

W (F )cd

F,SO(n)p

, for some 1< p2 andc >0, (1.4)

is assumed far fromSO(n). Similar results have been obtained in[11]assuming also a bound of orderpfrom above.

The reason for considering energies satisfying (1.4) without any bound from above is not purely academic. Indeed, for a large class of compressible rubber-like materials, (1.4) is the appropriate behaviour (see Remark2.8and the discussion in[1]for a multiwell case).

The main tool for the proof of the compactness of the minimizers considered in [5] is theGeometric Rigidity Lemmaof[8]. To obtain the same result when (1.3) holds only nearSO(n), while (1.4) holds far fromSO(n), we need a version with two exponents of the Geometric Rigidity Lemma, similar to those used in[3,10,11].

The proof of theΓ-convergence in the present paper has been renewed with respect to[5], and also with respect to the further improvements introduced in[12]. The main simplification relies on some arguments developed in[8]for the rigorous proof of dimension reduction results.

Moreover, the strong convergence inW^1,pof the minimizers is obtained by adapting to our techniques some ideas introduced in[12]for the case of multiwell energies satisfying the analog of (1.3). We hope that all our results can be extended to multiwell energies satisfying only (1.4) far from the wells.

2. Setting of the problem and main results

Throughout the paper,d(·,·)denotes the Euclidean distance both between two points and between a point and a set.

The space ofn×nreal matrices is identified withR^n×n;SO(n)is the set of rotations,Sym(n)andSkw(n)the sets of symmetric and skew-symmetric matrices, respectively, Psym(n) the set of positive definite symmetric matrices, Lin⁺(n) the set of invertible matrices with positive determinant. Given M∈R^n×n,symM andskwM denote the symmetric and the skew-symmetric part ofM, respectively.

The reference configuration Ω is a bounded connected open set of Rⁿ with Lipschitz boundary ∂Ω. We will prescribe a Dirichlet condition on a part ∂_DΩ of ∂Ω with Lipschitz boundary in ∂Ω, according to the following definition.

Definition 2.1.Let us define

Q:=(−1,1)ⁿ, Q⁺:=(−1,1)ⁿ⁻¹×(0,1),

Q₀:=(−1,1)ⁿ⁻¹× {0}, Q⁺₀ :=(−1,1)ⁿ⁻²×(0,1)× {0}.

We say that E⊆∂Ω has Lipschitz boundary in ∂Ω if it is nonempty and for everyx in the boundary ofE for the relative topology of∂Ω there exist an open neighbourhood U of x inRⁿ and a bi-Lipschitz homeomorphism ψ:U→Qsuch that

ψ (U∩Ω)=Q⁺, ψ (U∩∂Ω)=Q0, ψ (U∩E)=Q⁺₀.

The Sobolev spaceW^1,p(Ω,Rⁿ)will be denoted byW^1,p. To deal with the Dirichlet boundary condition, for every h∈W^1,pwe introduce the set

W_h^1,p:=

u∈W^1,p: u=hHⁿ⁻¹-a.e. on∂_DΩ

, (2.1)

where the equality on∂DΩrefers to the traces of the functions on the boundary∂Ω, andHⁿ⁻¹denotes the(n−1)- dimensional Hausdorff measure.

We suppose the material to be hyperelastic and that the stored energy densityW:Ω×R^n×n→ [0,∞]isL ×B- measurable, whereL andB are theσ-algebras of the Lebesgue measurable subsets of Rⁿ and Borel measurable subsets ofR^n×n, respectively. We assume thatW satisfies the following properties for a.e.x∈Ω:

(i) W (x,·)is frame indifferent;

(ii) W (x,·)is of classC² in some neighbourhood ofSO(n), independent ofx, where the second derivatives are bounded by a constant independent ofx;

(iii) W (x, F )=0 ifF ∈SO(n);

(iv) W (x, F )g_p(d(F,SO(n))), for some 1< p2, whereg_p: [0,∞)→Ris defined by g_p(t):=

t²

2, if 0t1,

t^p

p +¹₂−_p¹, ift >1. (2.2)

Observe that these assumptions are compatible with the conditionW (x, F )= ∞, if detF 0, which is classical in the context of finite elasticity. Also, observe thatgp is a convex function. In what followsD² denotes the second differential with respect to the variable F ∈R^n×n, so that D²W (x, F )[M]² means the second differential of the functionW with respect toF, evaluated at the point(x, F )and applied to the pair[M, M]. By frame indifference, for a.e.x∈Ωwe have that

D²W (x, I )[M]²=D²W (x, I )[symM]², for everyM∈R^n×n. (2.3) Together with assumption (iv), this implies that the quadratic formD²W (x, I )[·]²is null onSkw(n)and satisfies the coerciveness condition

D²W (x, I )[symM]²|symM|², for a.e.x∈Ωand everyM∈R^n×n. (2.4) Energy densities for which estimate (iv) holds only with 1< p <2 are commonly used in the study of compressible elastomers (see Remark2.8).

The load is modelled by a continuous linear functionalL :W^1,p→R. Ifv∈W^1,prepresents the deformation of the elastic body, the stable equilibria of the elastic body are obtained by minimizing the functional

Ω

W (x,∇v) dx−L(v),

under the prescribed boundary conditions. We are interested in the case where the load has the formεL and we want to study the behaviour of the solution asεtends to zero. We write

v=x+εu

and we assume Dirichlet boundary condition of the form v=x+εh Hⁿ⁻¹-a.e. on∂_DΩ,

with a prescribedh∈W^1,∞. The corresponding minimum problem forubecomes min

W^1,p Ω

W (x, I+ε∇u) dx−εL(εu)

, (2.5)

where the termεL(x)has been neglected since it does not depend onu. The following theorem is the main result of the paper. It describes the behaviour of the minimizers of (2.5).

Theorem 2.2. Assume that W :Ω ×R^n×n → [0,∞] satisfies conditions (i)–(iv) for some 1 < p2, and let h∈W^1,∞. For everyε >0let

m_ε:= inf

u∈W_h^1,p

1 ε²

Ω

W (x, I+ε∇u) dx−L(u)

, (2.6)

and let{u_ε}ε>0be a sequence such that 1

ε²

Ω

W (x, I+ε∇u_ε) dx−L(u_ε)=m_ε+o(1). (2.7)

Then,{u_ε}converges strongly inW^1,pto the unique solution of the problem m:= min

u∈W_h^1,2

1 2

Ω

D²W (x, I ) e(u)2

−L(u)

, (2.8)

wheree(u):=sym(∇u). Moreover,m_ε→m.

In the case 1< p <2, Theorem2.2asserts that a sequence of “almost minimizers” inW_h^1,p for theε-problems converges to a minimizer for the limit problem in a different Sobolev space: indeed, the limit problem is formulated inW_h^1,2.

In the case p=2, weak convergence of the “almost minimizers” has already been proved in[5]. Theorem2.2 extends this result to the case 1< p2 and provides also strong convergence. The proof is based on the following three results which are proved in Sections3,4, and5, respectively. These involve the functionalsFε,F :W^1,p→ [0,∞]defined by

Fε(u):=

₁

ε²

ΩW (x, I+ε∇u) dx, ifu∈W_h^1,p,

∞, otherwise, (2.9)

and

F(u):=

₁

2

ΩD²W (x, I )[e(u)]²dx, ifu∈W_h^1,2,

∞, otherwise, (2.10)

and the functionalsGε,G :W^1,p→(−∞,∞]defined by

Gε:=Fε−L, G:=F −L. (2.11)

Observe that, due to the growth property (iv) ofW, the functionalsGεandG are bounded from below.

Theorem 2.3.Assume thatW:Ω×R^n×n→ [0,∞]satisfies conditions(i)–(iv)for some1p2. There exists a constantC >0depending onΩ,∂DΩ, andpsuch that for everyh∈W^1,pand every sequence{uε} ⊆W_h^1,pwe have

Ω

|∇uε|^pdxC

1+Fε(uε)+

∂_DΩ

|h|dHⁿ⁻¹ 2

, (2.12)

for everyε >0sufficiently small.

The previous theorem ensures that, if {u_ε}is a sequence in W_h^1,p such that{Fε(u_ε)}is bounded, then {u_ε}is bounded inW^1,p, hence a subsequence converges weakly inW^1,p.

Theorem 2.4.Under the hypotheses of Theorem2.2, for everyε_j→0we have that Fεj

−→Γ F, asj → ∞, in the weak topology ofW^1,p.

Theorem2.4, together with the compactness result provided by Theorem2.3, implies the convergence of minima and the weak convergence of minimizers, using standard results onΓ-convergence. The next theorem and the previous remarks allow us to obtain the strong convergence of minimizers.

Theorem 2.5.Under the hypotheses of Theorem2.2, letεj→0and let{uj}be a recovery sequence foru∈W_h^1,2, that isu_j uweakly inW^1,pandFε_j(u_j)→F(u). Then{u_j}converges strongly inW^1,p.

Remark 2.6 (On the condition ∂_DΩ = ∅). Observe that in Theorem 2.3 the assumption ∂_DΩ = ∅ is crucial.

When ∂_DΩ = ∅, inequality (2.12) is false, as the following example shows. Consider the simple case W (F ):=

g_p(d(F,SO(n)))for everyF∈Lin⁺(n). For everyε >0 and someR∈SO(n)\ {I}, set uε(x):=R−I

ε x, x∈Ω.

In this case, we have that

Ω

|∇uε|^pdx=|Ω||R−I|^p

ε^p → ∞, asε→0⁺, whereas

Fε(u_ε)= 1 ε²

Ω

g_p d

I+ε∇u_ε,SO(n)

dx=0, for everyε >0.

Remark 2.7(On the conditionh∈W^1,∞).In Theorems2.2,2.4and2.5the hypothesish∈W^1,∞cannot be replaced byh∈W^1,2, unlessW satisfies suitable bounds from above, which are not natural in the context of finite elasticity.

Consider the simple case∂_DΩ=∂Ω,L =0, and assume that for somer >2 we have W (F )|F|^r for|F|large enough.

By well-known properties of the images of Sobolev spaces under the trace operator, there exists h∈W^1,2 such that

u∈W^1,r: u=hHⁿ⁻¹-a.e. on∂Ω

= ∅. (2.13)

Let us prove that Fε(u)= ∞ for every u∈W^1,p. Assume by contradiction that there exists u∈W^1,p with Fε(u) <∞. By (2.9) we have that ∇u∈L^r, hence u∈W^1,r, because Ω has Lipschitz boundary. This contra- dicts (2.13). Therefore{Fε}cannotΓ-converge toF, becauseF(h) <∞.

Remark 2.8(Model energy densities).A large class of models where the energy density grows quadratically near the wells and less than quadratically elsewhere is provided by rubber elasticity, when one wishes to take into account the compressibility of the material. We recall that we have formalized this growth behaviour by introducing, as bound from below of our energies, the function

g_p d

·,SO(3)

, for some 1< p <2,

whereg_pis the function defined in (2.2). For simplicity, we focus on the homogeneous case.

A common practice to pass from an incompressible model, with associated energy density W˜ defined on {F∈R^3×3: detF=1}, to a corresponding compressible modelW(see, e.g.,[2,7,9]) is to define

W (F ):= ˜W

(detF )^−1/3F

+W_vol(detF ), for everyF ∈Lin⁺(3), whereW_volis such that

W_vol0 and W_vol(t)=0 if and only if t=1.

For example, we can takeW_volof the form W_vol(t)=c

t²−1−2 logt

, for everyt >0,

forc >0. Consider first theNeo-Hookeanincompressible model for hyperelastic materials, where the energy density is of the form

W˜_N(F ):=a

|F|²−3

, for everyF∈R^3×3with detF =1,

for a certaina >0. Following the procedure described above, we consider the corresponding compressible energy density defined for everyF∈Lin⁺(3)by

W_N(F ):= ˜W_N F

(detF )^1/3

+Wvol(detF )

=a

|F|² (detF )^2/3−3

+Wvol(detF ).

Let us check thatW_N has “gp-growth”. By using the well-known inequality between arithmetic and geometric mean, it is easy to see that

W_N 0 and W_N(F )=0 if and only if F ∈SO(3). (2.14)

Moreover, recalling the Green–St. Venant strain tensorE=¹₂(F^TF −I )and using simple rules of tensor calculus, it turns out that in the small strain regime (that is, the regime of the deformation gradients which vary nearSO(3)), W has the expression

W_N(F )=μ|E|²+λ

2tr²E+o

|E|²

, (2.15)

where

μ=2a, λ=4

−a 3 +c

.

The parametersμandλ+²₃μhave the physical meaning of a shear modulus and a bulk modulus, respectively. Since

|E|²¹₃tr²Efor everyE∈Sym(3), from (2.15) we obtain that W_N(F )min{μ,6c}|E|²+o

|E|² , and in turn,

W_N(F )1

2min{μ,6c}|E|², (2.16)

if |E| is small enough, that is, ifd(F,SO(3))is small enough. Since|√

C−I||C−I|for everyC∈Psym(3), from (2.16) we obtain that

W_N(F )1

8min{μ,6c}F^TF−I²=1

8min{μ,6c}d²

F,SO(3)

, (2.17)

ifd(F,SO(3))is sufficiently small. Now, we want to study the growth ofW in the regime|F| → ∞. In this case, if detF is bounded, then

W_N(F )C|F|²−3aCd˜ ²

F,SO(3)

(detF bounded), (2.18)

for someC,C >˜ 0. In the case detF → ∞, we have that W_N(F )K

|F|²

det^2/3F +det²F

, for someK >0. By using Young’s inequality

xyx^p p +y^q

q 1

p+1 q =1

withx=(_detF^|F|3)^1/2andy=(detF )^1/2, it is easy to show that W_N(F )K|F|^3/2Kd˜ ^3/2

F,SO(3)

(detF → ∞), (2.19)

for someK >˜ 0. (2.14), (2.17), (2.18) and (2.19) show thatW_N hasg_p-growth from below withp=³₂. It is important to notice thatW_N has not quadratic growth everywhere. In particular,W_N has not quadratic growth in the regime detF → ∞. This can be checked by taking into account deformation gradients of the type

F =

⎡

⎣λ² 0 0 0 ¹_λ 0

0 0 1

⎤

⎦, withλ0. (2.20)

As a second example, we consider theMooney–Rivlincompressible model given, for somea,b >0, by W_M(F ):=a

|F|² (detF )^2/3−3

+b

(detF )^2/3F⁻¹²−3

+Wvol(detF )

=W_N(F )+b

(detF )^2/3F⁻¹²−3

, (2.21)

for everyF ∈Lin⁺(3), and derived from the corresponding incompressible version as explained before. The inequality between arithmetic and geometric mean implies that the second summand in (2.21) is nonnegative, so that, from (2.14), we have that

W_M0 and W_M(F )=0 if and only if F ∈SO(3).

The formula for the small strain regime is given by (2.15), with μ=2(a+b), λ=4

−a+b 3 +c

.

From the fact that W_N has g_p-growth and from the positiveness of the second summand of (2.21) theg_p-growth of W_M trivially follows. Also in this case, deformation gradients of the type (2.20) show thatW_M does not grow quadratically everywhere.

Finally, we mention someOgden-typecompressible energy densities:

W_O(F ):=

m i=1

a_i

tr((F^TF )^γi/2) (detF )^γi/3 −3

+W_vol(detF ),

defined for everyF ∈Lin⁺(3), for somem1 anda_i, γ_i>0,i=1, . . . , m. The formula forW_O in the small strain regime is again given by (2.15), with

μ=2 m i=1

ai, λ=4

−1 3

m i=1

ai+c

.

Arguing similarly to the Neo-Hookean and the Mooney–Rivlin models, we obtain thatW_O attains its minimum 0 atSO(3). By using Young’s inequality and proper counterexamples, it is possible to show thatW_O hasgp-growth for some 1< p <2 (p depending on the exponentsγ_i), but not a quadratic growth in general, if 0< γ_i <3 for every i=1, . . . , mandγ_i>⁶₅for at least one indexi∈ {1, . . . , m}.

3. Compactness

In this and in the next sections we give the proofs of the results stated in Section2. To simplify the exposition, the proofs are given only whenW does not depend explicitly onx. The proofs in the general case require only minor modifications.

The compactness result requires the following extension of the well-known geometric rigidity result of[8], where a power ofd(∇v,SO(n))is replaced byg_p(d(∇v,SO(n))).

Lemma 3.1(Geometric rigidity). Letg_p be the function defined in(2.2). There exists a constantC=C(Ω, p) >0 with the following property:for everyv∈W^1,pthere exists a constant rotationR∈SO(n)satisfying

Ω

g_p

|∇v−R| dxC

Ω

g_p d

∇v,SO(n)

dx. (3.1)

Similar versions of Lemma 3.1 can be found in [3,10,11]. For sake of completeness, we give the proof in Appendix A.

We need two more lemmas in order to prove Theorem2.3.

Lemma 3.2.LetS⊆Rⁿbe a boundedH^m-measurable set with0<H^m(S) <∞for somem >0. Then

|F|S:=min

ζ∈Rⁿ

S

|F x−ζ|dH^m

is a seminorm onR^n×n. Define S0:=

x∈S: H^m

S∩B_ρ(x)

>0for everyρ >0 ,

and letaff(S0)be the smallest affine space containingS₀. LetK⊆R^n×nbe a closed cone such that dim

Ker(F )

<dim aff(S0)

, for everyF ∈K\ {0}. (3.2)

Then, there exists a constantC=C(S) >0such that C|F||F|S, for everyF ∈K.

Proof. It is enough to repeat the proof of[5, Lemma 3.3], replacing theL²norm with theL¹norm. 2

We will use the next lemma also in the proof of the Γ-convergence result. In what follows and in the rest of the paper we denote byCa positive constant which may change from line to line.

Lemma 3.3.Letε >0andu_ε∈W_h^1,p. Under the hypotheses of Theorem2.3, letR_ε∈SO(n)be a constant rotation satisfying(3.1)withv=x+εu_ε. Then,

|I−Rε|²Cε²

Fε(uε)+

∂_DΩ

|h|dHⁿ⁻¹ 2

,

whereCdepends only onΩ,∂_DΩ, andp.

Proof. Consider the deformationv_ε:=x+εu_ε. Lemma3.1tells us that there exists a constant rotationR_ε∈SO(n) such that

Ω

g_p

|∇v_ε−R_ε| dxC

Ω

g_p d

∇v_ε,SO(n) dx,

whereCdepends only onΩandp. Then, by assumption (iv) onW, we have that

Ω

g_p

|∇v_ε−R_ε| dxC

Ω

W (∇v_ε) dx=Cε²Fε(u_ε).

Jensen inequality thus implies g_p

1

|Ω|

Ω

|∇v_ε−R_ε|dx

Cε²Fε(u_ε). (3.3)

Poincaré–Wirtinger inequality and the continuity of the trace operator give

∂DΩ

|v_ε−R_εx−ζ_ε|dHⁿ⁻¹C

Ω

|∇v_ε−R_ε|dx,

whereζ_ε:=_|Ω¹_|

Ω(v_ε−R_εx) dxandC depends onΩ, so that, sincev_ε=x+εhHⁿ⁻¹-a.e. on∂_DΩ, we obtain

∂DΩ

(I−R_ε)x−ζ_εdHⁿ⁻¹C

Ω

|∇v_ε−R_ε|dx+ε

∂DΩ

|h|dHⁿ⁻¹

. (3.4)

Now, let us use Lemma3.2withS=∂_DΩand withKequal to the closed cone generated byI−SO(n). Showing first that everyF∈Kbelongs to the cone generated byI−SO(n)or toSkw(n), it is easy to prove that everyF∈K\ {0}

is such that dim

Ker(F )

< n−1.

On the other hand,∂Ω Lipschitz implies that the right-hand side of (3.2) is equal ton−1. Thus, we can apply Lemma3.2to(I−R_ε)∈Kand write that

C|I−R_ε|min

ζ∈Rⁿ

∂DΩ

(I−R_ε)x−ζdHⁿ⁻¹, (3.5)

whereC depends on∂_DΩ and not onε. From (3.4) and (3.5) we obtain that

|I−R_ε|²C 1

|Ω|

Ω

|∇v_ε−R_ε|dx 2

+ε²

∂_DΩ

|h|dHⁿ⁻¹ 2

. (3.6)

We conclude the proof by distinguishing two cases. If

Ω|∇v_ε−R_ε|dx|Ω|, then (3.3) and the definition ofg_ptell us that

1 2

1

|Ω|

Ω

|∇v_ε−R_ε|dx 2

Cε²Fε(u_ε).

Using this last inequality in (3.6), it turns out (2.12). If

Ω|∇v_ε−R_ε|dx >|Ω|, again (3.3) and the definition ofg_p tell us that

Cε²Fε(uε) > 1 2.

This bound from below ofε²Fε(uε)gives trivially (2.12), in view of the fact that|I−Rε|2√ n. 2 For the proof of Theorem2.3we will need the following estimate

g_p(s+t )C

g_p(s)+t²

, for everys, t0, (3.7)

for a certainC depending onp. This estimate can be easily deduced from the convexity ofg_pand from the growth properties ofg_pwhich give

gp(t)1 pmin

t^p, t²

and gp(2t )Cgp(t), for everyt0, for someCdepending onp.

Proof of Theorem2.3. LetR_ε be given by Lemma3.1forv_ε:=x+εu_ε, for everyε >0. By using (3.7), we have that

Ω

g_p

|ε∇u_ε| dxC

Ω

g_p

|∇v_ε−R_ε|

+ |I−R_ε|² dx

C

Ω

g_p d

∇v_ε,SO(n)

dx+ |I−R_ε|²

,

where in the last inequality we have used Lemma3.1. Assumption (iv) onW and Lemma 3.3then imply that for someC, depending onΩ,∂DΩ, andp,

Ω

g_p

|ε∇u_ε|

dxCε²

Fε(u_ε)+

∂DΩ

|h|dHⁿ⁻¹ 2

. (3.8)

In particular, from (3.8) and from the definition ofg_pwe obtain

{x∈Ω:|ε∇uε(x)|1}

|ε∇u_ε|²dx2

Ω

g_p

|ε∇u_ε| dx

Cε²

Fε(u_ε)+

∂DΩ

|h|dHⁿ⁻¹ 2

,

so that, by Hölder inequality, it turns out

{x∈Ω:|ε∇uε(x)|1}

|ε∇u_ε|^pdx

{x∈Ω:|ε∇uε(x)|1}

|ε∇u_ε|²dx p/2

|Ω|^1−(p/2)

Cε^p

Fε(u_ε)+

∂_DΩ

|h|dHⁿ⁻¹ 2p/2

Cε^p

1+Fε(u_ε)+

∂_DΩ

|h|dHⁿ⁻¹ 2

. (3.9)

Note that in (3.9) we have used the fact that t^p/21+t, for everyt0.

On the other hand, from (A.2) and again from (3.8) we obtain that

{x∈Ω:|ε∇u_ε(x)|>1}

|ε∇uε|^pdxC

{x∈Ω:|ε∇u_ε(x)|>1}

gp

|ε∇uε| dx

Cε²

Fε(u_ε)+

∂DΩ

|h|dHⁿ⁻¹ 2

. (3.10)

Inequalities (3.9) and (3.10) imply that (2.12) holds. 2

In the next remark we construct a counterexample which shows that Theorem 2.3 is not true in general for p∈(0,1).

Remark 3.4.Letp∈(0,1)and consider the simple case in whichΩis the open unitary ballB(0,1)inR²,W (F ):=

g_p(d(F,SO(2)))for everyF ∈Lin⁺(2),h=0, andL =0. For anyε >0 and someα >0 to be chosen, we introduce the set

S_ε:=

x∈R²: 1

2<|x|<1 2+ε^α

.

For every ε >0 sufficiently small, Sε is an open annulus strictly included in Ω. We want to define a sequence {u_ε} ⊆W₀^1,psuch that the valuesFε(u_ε)are equibounded and

Ω|∇u_ε|^pdx→ ∞asε→0⁺. In order to do this, we consider for everyε >0 arbitrarily small a functionϕε∈C_c^∞(Ω,R)such that supp(ϕε)⊆B(0,¹₂)∪Sε, 0ϕε1, ϕ_ε≡1 onB(0,¹₂)and

|∇ϕ_ε| C

ε^α for someCindependent ofε. (3.11)

Then, we chooseR∈SO(2)\ {I}and define the function u_ε(x):=ϕ_ε(x)R−I

ε x, x∈Ω,

which belongs toC^∞for everyε >0 sufficiently small. Observe that

Ω

|∇u_ε|^pdx

B0,¹₂

|∇u_ε|^pdx=π|R−I|^p 4ε^p , so that

Ω|∇uε|^pdx→ ∞asε→0⁺(for every choice ofα >0). Now, let us compute

∇u_ε(x)=1 ε

ϕ_ε(x)(R−I )+

(R−I )x

⊗ ∇ϕ_ε(x)

and observe that∇u_ε≡0 onΩ\ B

0,¹₂

∪S_ε

, so thatd(I+ε∇u_ε,SO(2))≡0 on the same set. Thus, recalling that g_pis increasing, it turns out that

ε²Fε(u_ε)

B0,¹₂∪Sε

g_p

|I+ε∇u_ε−R| dx

Sε

g_p

|R−I|

1+ |x||∇ϕ_ε|

dx, (3.12)

where in the last inequality we have also used the fact thatϕ_ε≡1 onB(0,¹₂). Therefore, from (2.2) and (3.12) we obtain that

Fε(u_ε) C ε²

Sε

1+ |∇ϕ_ε|^p

dx, (3.13)

for someCindependent ofε. Using (3.11) and noticing that|Sε| =π ε^α+o(ε^α), (3.13) implies that Fε(u_ε) C

ε²

π ε^α+o ε^α

1+ 1 ε^αp

,

so that{Fε(uε)}turns out to be bounded wheneverα >₁₋²_p. We end this section with the following corollary.

Corollary 3.5. Under the hypotheses of Theorem 2.3, the functionals Gε are equicoercive in the weak topology ofW^1,p.

Proof. Lett∈Rand{uε}be a sequence withGε(uε)t, so that{uε} ⊆W_h^1,p. Thus, by the definition ofGε (2.11), we have

Fε(uε)t+L(uε).

Theorem2.3implies that for everyεsufficiently small

Ω

|∇uε|^pdxC

1+L(uε)+

∂_DΩ

|h|dHⁿ⁻¹ 2

,

for someCindependent ofε. By Poincaré inequality, this gives u_ε^p_W1,pC

u_εW^1,p +1

, (3.14)

whereCnow depends also onhandL. Therefore, sincep >1, from (3.14) we obtain thatu_εW^1,pis bounded. 2 Observe that the proofs of Theorem2.3, Lemma3.3and Corollary3.5do not use the fact that∂_DΩ has Lipschitz boundary in∂Ω(see Definition2.1): actually, these results hold under the weaker hypothesisHⁿ⁻¹(∂_DΩ) >0.

4. Γ-convergence

Consider a sequenceεj →0⁺asj→ ∞. By Theorem2.3, we can characterize theΓ-limit in the weak topology ofW^1,pin terms of weakly converging sequences (see[6, Proposition 8.10]). In particular, we have that

F(u):=Γ-lim inf

j→∞ Fε_j(u)=inf

lim inf

j→∞ Fε_j(u_j): u_j uweakly inW^1,p ;

F(u):=Γ-lim sup

j→∞ Fε_j(u)=inf

lim sup

j→∞ Fε_j(uj): uj uweakly inW^1,p

. (4.1)

Thus, in order to prove Theorem2.4, we will show thatF(u)F(u)andF(u)lim infj→∞Fε_j(u_j), for every u∈W^1,pand everyu_j uweakly inW^1,p.

Proof of Theorem2.4. (I) We want to show thatF(u)F(u). Consider the nontrivial caseF(u) <∞, so that u∈W_h^1,2and

F(u)=1 2

Ω

D²W (I ) e(u)2

dx.

Suppose firstu∈W^1,∞. The boundedness of∇uand assumption (ii) onW, together with the fact thatW (I )=0 and DW (I )=0, imply that

jlim→∞

1 ε_j²W

I+ε_j∇u(x)

=1

2D²W (I )

∇u(x)2

, for a.e.x∈Ω, and that there existsC >0 such that for everyεj>0 sufficiently small

W (I+ε_j∇u)ε²_jC|∇u|², a.e. inΩ.

Then, by dominated convergence and by (2.3), we obtain

jlim→∞

1 ε_j²

Ω

W (I+ε_j∇u) dx=1 2

Ω

D²W (I ) e(u)2

dx.

Therefore, by (4.1), F(u)= lim

j→∞Fεj(u)F(u). (4.2)

Consider now the general caseu∈W_h^1,2. Since∂_DΩhas Lipschitz boundary in∂Ω, from PropositionA.2we have that there exists a sequence{u_k} ⊆W_h^1,∞such thatu_k→ustrongly inW^1,2, ask→ ∞. Observe that by (4.2) we haveF(u_k)F(u_k)for everyk. Thus, by the weak lower semicontinuity ofFinW^1,pand the strong continuity ofF inW_h^1,2, it turns out that

F(u)= lim

k→∞F(u_k)lim inf

k→∞ F(u_k)F(u).

(II) We want to prove that, if u_j u weakly in W^1,p, then F(u)lim infjFεj(u_j). Consider the nontrivial case lim infj→∞Fεj(u_j) <∞so that, up to a subsequence, we can suppose{Fεj(u_j)}bounded and, in particular, {u_j} ⊆W_h^1,p. Let 1Bj be the characteristic function ofB_j, where

Bj:=

x∈Ω: ∇uj(x) 1

√ε_j

. (4.3)

Claim 1.We have that{1Bj∇u_j}is bounded inL².

Proof. By Lemma3.1and by the growth hypothesis onW we have that for everyj there existsR_j∈SO(n)such that

Ω

g_pI+ε_j∇u_j(x)−R_jdxε_j²CFε_j(u_j)Cε_j², (4.4)

where the last inequality follows from the boundedness of{Fεj(u_j)}. Considering the set A_j:=

x∈Ω: I+ε_j∇u_j(x)−R_j3√ n

,

it is easy to check thatB_j⊆A_j for everyj large enough, so that

B_j

|∇u_j|²dx 2 ε²_j

A_j

|ε_j∇u_j+I−R_j|²+ |I−R_j|²

dx. (4.5)

Therefore, by using (A.1) and the definition ofA_j, from (4.5) we obtain that

B_j

|∇u_j|²dx C ε²_j

A_j

g_p

|ε_j∇u_j+I−R_j|

+ |I−R_j|² dx

C

1+|I−R_j|² ε_j²

, (4.6)

where in the last inequality we have used (4.4) andC depends onΩ andp. Since{F(u_j)}is bounded, Lemma3.3 tells us that|I−R_j|²/ε²_jis bounded. This fact, together with (4.6), gives the claim. 2

Claim 2.∇u∈L²and, up to a subsequence, we have that 1Bj∇u_j∇u weakly inL².

Proof. By Claim 1, we have that, up to a subsequence,

1Bj∇u_j v weakly inL², (4.7)

for somev∈L². Let us prove that

1B_j^c∇u_j→0 strongly inL^α, (4.8)

for everyα∈ [1, p). We first observe that|B_j^c| →0, by Chebyshev inequality. Taking into account the boundedness of{u_j}inW^1,p, by Hölder inequality we obtain

Ω

|1B_j^c∇u_j|^αdx

Ω

|∇u_j|^pdx α/p

B_j^c(p−α)/pCB_j^c(p−α)/p→0, which proves (4.8).

The weak convergence ofu_j touinW^1,p implies also that∇u_j∇uweakly inL^α, for everyα∈ [1, p). This fact, together with (4.8), gives that

1B_j∇u_j=(∇u_j−1_B^c

j∇u_j) ∇u weakly inL^α, (4.9)

for everyα∈ [1, p). By (4.7) and (4.9) we conclude that∇u=v∈L²and Claim 2 follows. 2 From assumptions (ii) and (iii) onW it is easy to show that

W (I+F )1

2D²W (I )[F]²−η

|F|

|F|², for everyF ∈R^n×n,

whereηis an increasing function on[0,∞)such thatη(t )→0 ast→0⁺. Therefore, we can write