An approximation theorem for sequences of linear strains and its applications

ESAIM: Control, Optimisation and Calculus of Variations

April 2004, Vol. 10, 224–242 DOI: 10.1051/cocv:2004001

AN APPROXIMATION THEOREM FOR SEQUENCES OF LINEAR STRAINS AND ITS APPLICATIONS

Kewei Zhang

Abstract. We establish an approximation theorem for a sequence of linear elastic strains approaching a compact set inL¹ by the sequence of linear strains of mapping bounded in Sobolev spaceW^1,p. We apply this result to establish equalities for semiconvex envelopes for functions defined on linear strains viaa construction of quasiconvex functions with linear growth.

Mathematics Subject Classification. 26B25, 41A30, 49J45.

Received July 4, 2002. Revised September 22, 2003.

1. Introduction and main results

This paper establishes an approximation theorem for sequences of linear elastic strains approaching a compact set in the space of symmetric matrices. We apply the result to the study of equality of various semiconvex envelopes for functions defined on the space of linear strains. We show that under a simple coercivity condition, Q_e(f) =C(f) if and only ifR_e(f) =C(f), whereQ_e(f) and R_e(f) are the quasiconvex and rank-one convex envelopes off on linear strains respectively. Before we state our main results, let us introduce some notation.

ForA∈M^n×n – the space ofn×nreal matrices with the standard Euclidean inner product onRⁿ², we let e(A) = (A+A^T)/2, whereA^T is the transpose ofA. Ifuis a smooth mapping from a domain Ω⊂RⁿtoRⁿ, we calle(Du(x)) the linear elastic strain ofuwhereDuis the gradient ofu. LetM_sⁿ and (M_sⁿ)^⊥ be the subspaces of symmetric and skew-symmetric matrices in M^n×n respectively, we see thate(A) = P_Mn

s(A), whereP_Mn s is the orthogonal projection fromM^n×nontoM_sⁿ. Let dist(Y, K) = inf_X∈K|Y−X|be the distance function from Y ∈M_sⁿ to a closed setK⊂M_sⁿ. The following is our approximation theorem.

Theroem 1. LetΩ⊂Rⁿ be a Lipschitz domain andu_j∈C₀^∞(Ω,Rⁿ)such that

j→∞lim

Ωdist(e(Du_j(x)),B¯_R(0))dx= 0, (1.1)

where B¯_R(0) is the closed ball in M_sⁿ centered at 0 with radius R. Then there is a subsequence (u_jk) of (u_j), and a sequence of Lipschitz mappings v_k :Rⁿ→Rⁿ such that for every 1< p <∞,

Rⁿ|e(Dv_k)|^pdx≤C(n, p)<+∞, and lim

k→∞

Ω|e(Du_jk)−e(Dv_k)|dx= 0. (1.2)

Keywords and phrases. Linear strains, maximal function, approximate sequences, quasiconvex envelope, quasiconvex hull.

1 School of Mathematical Sciences, University of Sussex Brighton, BN1 9QH, UK; e-mail:[email protected]

c EDP Sciences, SMAI 2004

Note to Theorem 1. As kindly pointed to me by I. Fonseca and the referee, there is an alternative proof of Theorem 1 by usingA-qausiconvexity. Further details can be found in the remark after the proof of Theorem 1.

Corollary 1. Let (v_k)be the sequence of Lipschitz mappings given by Theorem 1. Then for each 1< p <∞, there are a skew-symmetric matrixA(p)_k ∈(M_sⁿ)^⊥ and some x_k ∈Rⁿ satisfying

ΩA(p)_k(x−x_k)dx= 0 such that the sequencew_k: Ω→Rⁿ defined by

w_k(x) =v_k− 1 meas(Ω)

Ωv_k(y)dy−A_k(p)(x−x_k), x∈Ω is bounded inW^1,p(Ω,Rⁿ),

Ω|Dw_k|^pdx≤C

Ω|e(Dw_k)|^pdx≤C₀(p)<+∞,

Ωw_kdx= 0, (1.3)

whereC,C₀(p)are positive constants independent ofk and

k→∞lim

Ω|e(Du_jk)−e(Dw_k)|dx= 0. (1.4) In Theorem 1, the ball ¯B_R(0) can be replaced by any compact set K ⊂M_sⁿ. Theorem 1 and Corollary 1 are motivated from an approximation result in [38] for W₀^1,1 approximating sequences of gradients approaching a compact set in M^N×n by a bounded W^1,∞ sequence. The statements of Theorem 1 and Corollary 1 are almost optimal in the sense that they are false if p = +∞ [24]. The main tool for establishing Theorem 1 is a generalized version of Liu’s Luzin type theorem [25] to the space of bounded deformations BD(Ω) [12].

Combining the result in [12] with some classical estimates for standard singular integral operators [30] enables us to prove Theorem 1. Corollary 1 then follows from Poincar´e’s inequality and Korn’s inequality [20].

The following is our main application of Theorem 1 and Corollary 1 to equalities of some semiconvex en- velopes for functions defined on linear strains. The study of quasiconvex functions defined on linear elastic strains is closely related to the variational approach to material microstructure by using geometrically linear models [9, 10, 18]. Functions defined onM_sⁿwith linear growth are important in the theory of plasticity [3,21,35].

For a continuous function f :M_sⁿ →Rbounded below, letQ_e(f) and R_e(f) be the quasiconvex and rank-one convex envelopes off respectively (see Sect. 2 for definitions). We denote byC(f) the convex envelope off. Theroem 2. Supposef :M_sⁿ→Ris continuous and satisfies, for A∈M_sⁿ

|A|→+∞lim f(A)

|A| = +∞. (1.5)

Then Q_e(f) =C(f) if and only ifR_e(f) =C(f).

Theorem 2 was established for functions f : M^N×n → R [43] under the coercivity condition (1.5) for A∈ M^N×n. The difference between Theorem 2 and that in [43] is that in the present situation, the function X → f(e(X)) is not coercive in the sense of (1.5) for X ∈ M^n×n. A weaker version of Theorem 2 was proved in [44], via an elementary argument for functions defined on linear strains under the assumption that f(A)≥c(|A|²−1).

The following result on the construction of quasiconvex functions on linear strains with linear growth will be used indirectly to establish Theorem 2 through some of its implications. The construction itself is of independent interest and its proof depends on Theorem 1 and Corollary 1.

Theroem 3. SupposeF : M_sⁿ → Ris quasiconvex on linear strains and is bounded below. Assume that 0≤ F(Y)≤C₀(1 +|Y|^p)for Y ∈M_sⁿ, and that for some realα≥0, the sub-level setK_α:={Y ∈M_sⁿ:F(Y)≤α}

is compact. Then, for every1≤p <+∞, the quasiconvex function on linear strains Q_edist^p(A, K_α),A∈M_sⁿ satisfies

C₀|Y|^p−C₁≤Q_edist^p(Y, K_α)≤C₂(1 +|Y|^p) (1.6) for Y ∈M_sⁿ and

K_α={Y ∈M_sⁿ:Q_edist^p(Y, K_α) = 0}, (1.7) whereC₀,C₁,C₂ are positive constants andQ_edist^p(Y, K_α)is the quasiconvex envelope on linear strains of the p-distance function dist^p(Y, K_α).

We need the following results on various semiconvex hulls of linear strains for compact sets inM_sⁿand their properties to establish Theorem 2. For theM^N×n version of these results, see [33, 37, 39]. We define two types of quasiconvex hulls on linear strains for closed subsets ofM_sⁿ.

Definition 1.1. LetK⊂M_sⁿ be closed, for 1≤p <∞, we defined the strong p-quasiconvex hull Q^e_p(K) and weakp-quasiconvex hullQ^e_p(K) respectively as

Q^e_p(K) =

X ∈M_sⁿ, f(A)≤ sup

Y∈Kf(Y), f quasiconvex on linear strains , 0≤f(A)≤C_f(1 +|A|^p)

· (1.8)

Q^e_p(K) ={A∈M_sⁿ, Q_edist^p(A, K) = 0};

and the strongp-rank-one convex hull of linear strains as R^e_p(K) =

X ∈M_sⁿ, f(A)≤ sup

Y∈Kf(Y), f rank-one convex on linear strains, 0≤f(A)≤C_f(1 +|A|^p)

· (1.9) Clearly, one hasR^e_p(K)⊂Q^e_p(K)⊂Q^e_p(K),R^e_p(K)⊂R^e_q(K) and Q^e_p(K)⊂Q^e_q(K) for 1≤q ≤p <∞. We can also show thatQ^e_p(K)⊂Q^e_q(K) (see Proof of Th. 3 and (4.1)).

In order to state Theorem 5, we need to define the so-called closed lamination convex hullL^e_c(K) on linear strains for a compact setK⊂M_sⁿ as follows [44].

Notice that the subspace (M_sⁿ)^⊥ of skew-symmetric matrices does not have rank-one matrices. We say that A ∈ M_sⁿ is a compatible linear strain if span[A]⊕(M_sⁿ)^⊥ has rank-one matrices. For example, the identity matrixI∈M_sⁿ is incompatible. Two linear strainsA, B∈M_sⁿ are called compatible ifA−B is a compatible linear strain and we call{A, B} a compatible pair. Clearly, A ∈M_sⁿ is compatible if and only if {0, A} is a compatible pair. This definition of compatibility is equivalent to that in [22], that is,A, B∈M_sⁿare compatible if eitherA−B is a rank-one matrix or A−B is of rank two and the two non-zero eigenvalues have opposite signs.

A set K ⊂M_sⁿ is called lamination convex on linear strains if for every compatible pair {A, B} ⊂ K, one has{tA+ (1−t)B,0≤t≤1} ⊂K. For a compact setK⊂M_sⁿ, the closed lamination convex hull on linear strainsL^e_c(K) is the smallest closed lamination convex set on linear strains that containsK.

The closed laminated convex hull for a compact setK⊂M^N×n was defined in [40] motivated from [28]. It is easy to see thatK⊂L^e_c(K)⊂R^e_p(K)⊂Q^e_p(K)⊂C(K).

Theroem 4. Let K⊂M_sⁿ be non-empty and compact. Then1≤p <∞,

Q^e_p(K) =Q^e_p(K) =Q^e₁(K). (1.10)

The following result is an implication of the results established in [44]. For the convenience of the reader, we give a proof in the Appendix.

Theroem 5. LetK⊂M_sⁿ be compact. ThenL^e_c(K) =C(K)if and only if Q^e₂(K) =C(K).

From Theorem 4 we see that we may replaceQ^e₂ by anyQ^e_p for 1≤p <∞. AlsoR^e₂(K) =C(K) if and only ifQ^e₂(K) =C(K).

In Section 2, we give notation and preliminaries which are needed for establishing our main results. We prove Theorem 1 and Corollary 1 in Section 3. In Section 4 we establish Theorems 2–4 by applying Theorem 1 directly or indirectly followed by some examples. In the Appendix we give a proof of Theorem 5 and establish an elementary result Lemma 4.1 which is needed in the proof of Theorem 3.

2. Notation and preliminaries

Throughout the rest of this paper Ω denotes a bounded open subset ofRⁿ.We denote by M^N×n the space of realN ×n matrices, with norm|P|= (trP^TP)^1/2. In this paper we are mainly interested in the case when N =n ≥2. We letC₀^∞(Ω,Rⁿ) be the space of smooth functions φ: Ω →Rⁿ having compact support in Ω.

We denote the Lebesgue spacesL^p(Ω,Rⁿ) and Sobolev spaces W^1,p(Ω,Rⁿ) andW₀^1,p(Ω,Rⁿ) for vector-valued functions u : Ω → Rⁿ as usual [2]. As in Section 1, we let M_sⁿ and (M_sⁿ)^⊥ be the subspaces of symmetric and skew-symmetric matrices respectively. We see that these two subspaces are orthogonal to each other. For X ∈M^n×n, we lete(X) = (X+X^T)/2 =P_Msⁿ(X) whereP_Msⁿis the orthogonal projection fromM^n×n toM_sⁿ. We denote weak convergence of sequences by. The Lebesgue measure of a measurable setS inRⁿis meas(S) while the complement of a setS⊂Rⁿ isS^c. We use variousC’s to denote positive constants such asC(n, p). In later sections, twoC(n, p)’s in the same line may not be the same. They just mean positive constants depending only onnandp.

We define the p-distance function fromY ∈M_sⁿ to a set K ⊂M_sⁿ by dist^p(Y, K) := inf_A∈K|Y −A|^p.The following are some results we need later.

Definition 2.1. (See [4, 26].) A continuous functionf :M^N×n→Ris quasiconvex if

Uf(P+Dφ(x)) dx≥f(P) meas(U)

for everyP ∈M^N×n,φ∈C₀^∞(U;R^N), and every bounded open subsetU ⊂Rⁿ . A functionf :M^N×n→Ris rank-one convex if for anyA, B∈M^N×n withB a rank-one matrix, the functiont→f(A+tB) is convex.

It is well-known now that quasiconvexity implies rank-one convexity [4, 11, 26] while the converse is not true [32]. To construct quasiconvex functions, we need the following

Definition 2.2. (See [11].) Suppose f : M^N×n → R is a continuous function. The quasiconvex envelope (rank-one convex envelope, respectively)Q(f) (R(f) respectively) off is defined by

Q(f) = sup{g≤f;gquasiconvex}, R(f) = sup{g≤f;g rank-one convex}·

It is well-known [11] that C(f)≤Q(f)≤R(f)≤f and the quasiconvex envelopeQf can be calculated by Qf(P) = inf

φ∈C₀^∞(Ω;R^N)

1 meas(Ω)

Ωf(P+Dφ(x)) dx, (2.1) where Ω⊂Rⁿ is a bounded domain. In particular the infimum in (2.1) is independent of the choice of Ω.

For a continuous function f : M_sⁿ → R, we say that f is quasiconvex (rank-one convex respectively) on linear strains, if the function F : M^n×n → Rdefined by F(X) = f(e(X)) is a quasiconvex (rank-one convex respectively) function.

We define the quasiconvex envelopeQ_e(f) and rank-one convex envelopeR_e(f) on linear strains for a con- tinuous functionf :M_sⁿ→Rby

Q_e(f) = sup{g≤f; gquasiconvex on linear strains}

R_e(f) = sup{g≤, f;grank-one convex on linear strains}·

The following simple statement is easy to prove.

Proposition 2.1. For a continuous function f : M_sⁿ → R, let F(X) = f(e(X)) for X ∈ M^n×n. Then Q(F(X)) =Q_e(f(e(X))).

Proof. ClearlyQ_e(f(e(X)))≤Q(F(X)) = sup{g≤f;g quasiconvex}.However, if we letD⊂Rⁿ be the unit cube, then

Q(F(X)) = inf

φ∈C₀^∞(D;Rⁿ)

DF(X+Dφ)dx= inf

φ∈C₀^∞(D;Rⁿ)

Df(e(X+Dφ))dx:=h(e(X)),

depending only one(X). Note thatX →h(e(X)) =Q(F(X)) is quasiconvex, hencehis quasiconvex on linear

strains andh≤f. By definition,h≤Q_e(f). The proof is finished.

We will use the following theorem concerning the existence and properties of Young measures [5, 19, 34].

Proposition 2.2. Let (z_j) be a bounded sequence in L¹(Ω;R^s). Then there exist a subsequence (z_jk) of (z_j) and a family (ν_x)_x∈Ω of probability measures onR^s, depending measurably onx∈Ω, such that

f(z_jk)

R^sf(λ)dν_x(λ), in L¹(Ω)

for every continuous functionf :R^s→Rsuch that(f(z_jk))is sequentially weakly relatively compact inL¹(Ω).

If the sequence z_j is in the form z_j = Du_j, where Ω ⊂ Rⁿ is open and bounded, and (u_j) is a bounded sequence inW^1,p(Ω,R^N) for some 1< p≤ ∞, then the corresponding Young measureν_x is calledp-gradient Young measures (see [10, 19, 23]). The Young measure istrivialifν_xis a Dirac measure fora.e. x. In this case there exists a function u such that ν_x is the Dirac measure atDu(x), and up to a subsequence, Du_k → Du almost everywhere.

One of the restrictions ofp-gradient Young measures [19] is that for every quasiconvex functionf :M^N×n→ R, satisfying|f(X)| ≤C(1 +|X|^p) when 1< p <∞,

M^N×n

f(λ)dν_x≥f

M^N×n

λdν_x

(2.2) for almost everyx∈Ω, (see for example, [8, 10, 19]).

Forr >0 andx∈Rⁿ,letB_r(x) ={y∈Rⁿ :|y−x|< r}and meas(B_r(x)) =ω_nrⁿ, we have ([30]), Definition 2.3. (The Maximal Function) Letf∈L¹_loc(Rⁿ), we define

(M f)(x) = sup

r>0

1 ω_nrⁿ

Br(x)|f(y)|dy, whereω_n is the volume of thendimensional unit ball.

We have the following weak-(1,1) and strong-(p, p) estimates [30][p. 5, Th. 1(b)]:

Proposition 2.3. If f ∈L¹(Rⁿ), then for everyλ >0,

meas({x∈Rⁿ: (M f)(x)> λ})≤ C(n) λ

Rⁿ|f|dx.

Iff ∈L^p(Rⁿ)for 1< p <∞, there is a constantC(n, p)>0 such that (M f)_L^p_(Rⁿ₎≤C(n)f_L^p_(Rⁿ₎, which also implies the weak-(p, p) estimate

meas({x∈Rⁿ : (M f)(x)> λ})≤C(n, p) λ^p

Rⁿ|f|^pdx.

The following results on convolution operators can be found in ([30], Ch. II., Ths. 3 and 4)

Proposition 2.4. Let K : Rⁿ → R be a 0-homogeneous function, smooth with mean value zero on the unit sphereSⁿ⁻¹. Then forf ∈L^p(Rⁿ),1≤p <∞, define

T(f)(x) =

|y−x|≥

K(y−x)

|y−x|ⁿ f(y)dy, >0. Then (a) there exists a constant A_p>0 independent of >0 so that

T(f)_L^p_(Rⁿ₎≤A_pf_L^p_(Rⁿ₎, 1< p <∞;

(b) lim_→0T(f) =T(f)exists inL^p(Rⁿ)norm (1< p <∞) while lim_→0T(f)(x)exists for almost every x∈Rⁿ when1≤p <∞ and

T(f)_L^p_(Rⁿ₎≤A_pf_L^p_(Rⁿ₎, 1< p <∞;

(c) let T^∗(f)(x) = sup_>0|T(f)(x)|. If f ∈L¹(Rⁿ), then the mapping f →T^∗(f) is of weak type-(1,1), that is,

meas({x∈Rⁿ: (T^∗f)(x)> λ})≤C(n) λ

Rⁿ|f|dx, for allλ >0; (2.3) (d) if 1< p <∞, then T^∗(f)_L^p_(Rⁿ₎≤A_pf_L^p_(Rⁿ₎,which implies the weak-(p, p) estimate

meas({x∈Rⁿ : (T^∗f)(x)> λ})≤C(n, p) λ^p

Rⁿ|f|^pdx. (2.4)

The following are some useful estimates for functions in the space of bounded deformations BD(Ω) and BD(Rⁿ) [3, 12, 21]. To simplify the statements, we only state the results for functions inW^1,1which is contained in BD.

LetRbe the class of rigid motions inRⁿ, that is, affine functions of the formAx+dwithAskew-symmetric n×nmatrix and d∈Rⁿ. The following Poincar´e type inequality is in [21] for functions inBD(Ω), however, we only consider functions inW^1,1(Ω,Rⁿ)⊂BD(Ω).

Proposition 2.5. LetΩ⊂Rⁿbe a bounded, connected open set with Lipschitz boundary and letR:BD(Ω)→ R be a continuous linear mapping which leaves the elements ofRfixed. Then there exists a constantC(Ω, R)>0

such that

Ω|u−R(u)|dx≤C(Ω, R)

Ω|e(Du)|dx, for allu∈W^1,1(Ω,Rⁿ).

When Ω⊂Rⁿ is an open ball, there is a precise representation of the rigid motionR(u) given by the following result [3] (we still state it forW^1,1functions).

Proposition 2.6. Let u ∈W^1,1(Rⁿ,Rⁿ), x∈ Rⁿ and > 0. Then there exist a vector d(e(Du))(x) and a skew-symmetric matrix A(e(Du))(x), such that

B(x)|u(y)−A(e(Du))(x)(y−x)−d(e(Du))(x)|dy≤C(n)

B(x)|e(Du(y))|dy, whereC(n)>0is a constant. Furthermore d(·)andA(·)can be represented as singular integrals

dⁱ(F)(x) = n l,m=1

|y−x|≥

Λⁱ_lm(y−x)

nw_n|y−x|ⁿF_lm(y)dy, A^ij (F)(x) =

n l,m=1

|y−x|≥

Γ^ij_lm(y−x)

2w_n|y−x|ⁿ⁺²F_lm(y)dy,

where F ∈L^p(Rⁿ, M_sⁿ), with 1≤p <∞, Λ andΓ are third and forth order smooth tensors with zero average on Sⁿ⁻¹, andΛ is0-homogeneous andΓ 2-homogeneous respectively.

If we define A^∗(F)(x) = sup_>0|A(F)(x)|, we see from Proposition 2.5 that A^∗ have the weak-(1,1), strong-(p, p) and weak-(p, p) estimates for F∈L^p(Rⁿ, M_sⁿ), 1≤p <∞.

The following is a simple variation of ([12], Th. 3.1) – the Luzin type theorem for BD functions. In the original statement in [12], it was stated for the caseλ=τ with our notation. However, by examining the proof, it is easy to see that the following can be deduced by using the original proof. We also notice thatS_u =∅in our setting, whereS_u is the singular part ofu(see [3]).

Proposition 2.7. Let u∈C₀^∞(Rⁿ,Rⁿ). We define for any λ >0 andτ >0

E^λ={x∈Rⁿ, M(e(Du))(x)< λ}, H^τ={x∈Rⁿ, A^∗(e(Du))(x)< τ}, whereA^∗ is defined as above. Then there is a Lipschitz mapping v_λ,τ :Rⁿ →Rⁿ such that

|v_λ,τ(x)−v_λ,τ(y)| ≤C(n)(λ+τ)|x−y|, x, y∈Rⁿ such that u(x) =v_λ,τ(x)on W_λ,τ =E^λ∩H^τ.

We conclude this section by stating a special form of Korn’s inequality ([20], Th. 8):

Proposition 2.8. Let Ω⊂Rⁿ be as in Proposition 2.5 and letu∈W^1,p(Ω, Rⁿ), 1< p <∞. Then there is a skew-symmetric matrix A_u such that

Ω|Du(x)−A_u|^pdx≤C

Ω|e(Du(x))|^pdx, whereC >0 is a constant independent of uandA_u.

3. Proof of Theorem 1 and Corollary 1

We prove Theorem 1 through Lemma 3.1 to Lemma 3.5.

Proof of Theorem 1. The idea of the proof is the following. We use Lemma 3.1 and Lemma 3.2 to show that e(Du_j) is “essentially” bounded in L^∞. Then for a carefully chosen subsequenceu_jk we use the Luzin type theorem for BD functions to obtain a sequence (v_k) of Lipschitz mappings such that v_k(x) = u_jk(x)

on a large subset W_jk of Ω while the Lipschitz constants of (v_k) are unbounded (Lem. 3.3). Then we show (Lem. 3.4) that the linear strains e(Dv_k) of these Lipschitz functions are in fact, bounded in L^p by using the estimates on A^∗ defined in Proposition 2.6 and the maximal function of e(Du_j). Finally we prove that lim_k→∞

Ω|e(Du_jk−Dv_k)|= 0 (Lem. 3.5).

Lemma 3.1. Under the assumptions of Theorem 1 we have

j→∞lim

{x∈Ω,|e(Duj(x))|≥2R}|e(Du_j(x))|dx= 0, (3.1) and|e(Du_j(x))|is equi-integrable on Ω.

Proof. Since lim_j→∞

Ωdist(e(Du_j(x)),B¯_R(0))dx= 0, we see that 0 = lim

j→∞

Ωdist(e(Du_j(x)),B¯_R(0))dx

≥lim sup

j→∞

{x∈Ω,|e(Duj(x))|≥2R}dist(e(Du_j(x)),B¯_R(0))dx

≥lim sup

j→∞ Rmeas ({x∈Ω,|e(Du_j(x))| ≥2R}),

where we have used the fact that when |e(Du_j(x))| ≥2R, dist(e(Du_j(x)),B¯_R(0))≥R. Next since we have dist(e(Du_j(x)),B¯_R(0))≥ |e(Du_j(x))| −R, (3.2) whenever|e(Du_j(x))|> R, we see that

0 = lim

j→∞

Ωdist(e(Du_j(x)),B¯_R(0))dx

≥lim sup

j→∞

{x∈Ω,|e(Duj(x))|≥2R}dist(e(Du_j(x)),B¯_R(0))dx

≥lim sup

j→∞

{x∈Ω,|e(Duj(x))|≥2R}(|e(Du_j(x))| −R) dx,

hence the first conclusion follows. The second claim is easy to prove because ¯B_R(0) is compact. In fact, if we letb_j=

Ωdist(e(Du_j(x)),B¯_R(0))dx,thenb_j →0 asj→ ∞. For any measurable subsetGof Ω, we have, from (3.2) that

b_j ≥

Gdist(e(Du_j(x)),B¯_R(0))dx≥

G|e(Duj(x))|dx−Rmeas(G) so that

G|e(Du_j(x))|dx≤b_j+Rmeas(G). The equi-integrability of|e(Du_j(x))|on Ω then follows easily from this inequality. In fact, for any >0, if we first chooseδ₁=/(2R), then there is someN >0, such thatb_j< /2 and

G|e(Du_j(x))|dx≤wheneverj > N and meas(G)< δ₁. Let δ₂ >0 be such that

G|e(Du_j(x))|dx≤ when meas(G)< δ₂and 1≤j≤N. Now if we takeδ= min{δ₁, δ₂}, the second conclusion then follows.

Now, if

a_j=

{x∈Ω,|e(Duj(x))|≥2R}|e(Du_j(x))|dx, (3.3) we see that a_j→0 asj→ ∞which follows from the proof of Lemma 3.1.

Now we extendu_jto be defined onRⁿby zero, then we may consider the maximal functionM(|e(Du_j)|) (x) of|e(Du_j(x))|. We have

Lemma 3.2. Forλ >2R,let

E_j^λ={x∈Rⁿ, M(|e(Duj)|) (x)< λ}, hence

E_j^{λ c}={x∈Rⁿ, M(|e(Duj)|) (x)≥λ}·

Let a_j be as defined in (3.3), then meas

E_j^{λ c}

≤ C(n)a_j

λ−2R →0 as j→ ∞. (3.4)

Proof. The proof of Lemma 3.2 is very similar to that of ([38], Lem. 3.1). We define h(t) =

0, if, |t| ≤2R,

|t| −2R, if, |t| ≥2R.

Thenh:R→R+ is a continuous function. We claim that

{x∈Rⁿ, M(|e(Du_j)|) (x)≥λ} ⊂ {x∈Rⁿ, M(h(|e(Du_j)|)) (x)≥λ−2R}· (3.5) We prove (3.5) as follows. IfM(|e(Duj)|) (x)≥λ, by definition, there is a sequence of positive numbers_k >0 andr_k >0 with_k→0 ask→ ∞such that

1 meas(B_rk(x))

B_rk(x)|e(Duj(y))|dy≥λ−_k. Since

M(h(|e(Du_j)|)) (x)≥ 1 meas(B_rk(x))

B_rk(x)h(|e(Du_j(y))|)dy

= 1

meas(B_rk(x))

{y∈B_rk(x),|e(Duj(y))|≥2R}(|e(Du_j(y))| −2R) dy

= 1

meas(B_rk(x))

B_rk(x)|e(Du_j(y))|dy− 1 meas(B_rk(x))

{y∈B_rk(x),|e(Duj(y))|≥2R}

2Rdy

− 1

meas(B_rk(x))

{y∈B_rk(x),|e(Duj(y))|<2R}|e(Du_j(y))|dy≥λ−_k−2R.

Passing to the limitk→ ∞we obtainM(h(|e(Duj))|) (x)≥λ−2R.Thus (3.5) is proved.

Now from the weak-(1,1) estimate of the maximal function (Prop. 2.4), we have meas ({x∈Rⁿ, M(h(|e(Du_j)|)) (x)≥λ−2R})≤ C(n)

λ−2R

Rⁿ

h(|e(Du_j(y))|)dy

≤ C(n) λ−2R

{y∈Ω,|e(Duj(y))|≥2R}|e(Du_j(y))|dy= C(n)

λ−2Ra_j→0, as j→ ∞, wherea_j is defined by (3.3). The proof is finished.

Since the sequence (a_j) defined by (3.3) converges to zero asj → ∞, we may find a subsequence (ajk) such that

a_jk ≤e^−(k+1). (3.6)

Recall the operator A^∗(e(Du)) defined following Proposition 2.6. Now we apply Proposition 2.7 to our se- quence u_jk.

Lemma 3.3. Let λ= 4R in Lemma 3.2 and let τ_k = 4kR, fork= 1,2, . . . Let H_jk={x∈Rⁿ, A^∗(e(Du_jk))(x)< τ_k}·

Then u_jk is a Lipschitz mapping on the set

W_jk=E_j^λ_k∩H_jk ={x∈RⁿM(|e(Du_j)|) (x)< λ, A^∗(e(Du_jk))(x)< τ_k}, (3.7) satisfying

|u_jk(x)−u_jk(y)| ≤C(n)(λ+τ_k)|x−y|=C(n)4(1 +k)R|x−y|. (3.8) From Lemma 3.3 and Kirszbraun’s theorem [36], there is a Lipschitz extensionv_k ofu_jk toRⁿ such that

|vk(x)−v_k(y)| ≤C(n)4(1 +k)R|x−y|, (3.9) for allx, y∈Rⁿ and

|Dvk(x)| ≤C(n)4(1 +k)R, (3.10)

for almost everyx∈Rⁿ andDv_k(x) =Du_jk(x) almost everywhere onW_jk ([16], Lem. 7.7).

Lemma 3.4. There is a constant C₀>0 independent ofv_k such that

Rⁿ|e(Dv_k(x))|^pdx≤C₀. Proof. For a measurable setS⊂Rⁿ, we let

J_p(w, S) =

S|e(Dw(x))|^pdx, 1≤p <∞ (3.11) as long as the right hand side of (3.11) is finite. We then have

Rⁿ|e(Dv_k(x))|^pdx=J_p(v_k,Rⁿ) =J_p(v_k, W_jk) +J_p(v_k,(W_jk)^c).

We see that

J_p(v_k, W_jk) =J_p(u_jk, W_jk)≤λ^pmeas(Ω) = (4R)^pmeas(Ω).

This follows from the fact that Dv_k=Du_jk almost everywhere onW_jk,u_jk is supported in Ω,W_jk ⊂E_j^λ_k and

|e(Dujk(x))| ≤M(|e(Dujk)|)(x)≤λ= 4R onE_j^λ_k. We also have

J_p(v_k,(W_jk)^c) =J_p(v_k,(H_jk∩E_j^λ_k)^c)≤J_p(v_k,(H_jk)^c∪(E_j^λ_k)^c)≤J_p(v_k,(H_jk)^c) +J_p(v_k,(E_j^λ_k)^c).

Notice that|Dvk(x)| ≤C(n)4R(k+ 1) andλ= 4R, we have

J_p(v_k,(E_j^λ_k)^c)≤[C(n)4R(k+ 1)]^pmeas((E^λ_jk)^c)≤C(n, p)R^p(k+ 1)^p (2R)^p a_jk

≤ C(n, p)R^p(k+ 1)^p

(2R)^p e^−(k+1)≤C(n, p)(k+ 1)^pe^−(k+1)≤C(n, p). (3.12)

To estimateJ_p(v_k,(H_jk)^c), we writee(Du_jk(x)) =e₁(Du_jk(x)) +e₂(Du_jk(x)),where e₁(Du_jk(x)) =

e(Du_jk(x)), if|e(Du_jk(x))|<4R, 0, if|e(Du_jk(x))| ≥4R;

e₂(Du_jk(x)) =

e(Du_jk(x)), if|e(Du_jk(x))| ≥4R, 0, if|e(Du_jk(x))|<4R.

Then from

A^∗(e(Du_jk))(x) =A^∗(e₁(Du_jk) +e₂(Du_jk))(x)≤A^∗(e₁(Du_jk))(x) +A^∗(e₂(Du_jk))(x), we have

(H_jk)^c ⊂ {x∈Rⁿ, A^∗(e₁(Du_jk))(x)≥τ_k/2} ∪ {x∈Rⁿ, A^∗(e₂(Du_jk))(x)≥τ_k/2}·

SinceDu_jk is supported in Ω ande₁(Du_jk)(x) is a bounded sequence in L^∞(Rⁿ), we have from the weak-(p, p) estimate of operatorA^∗(Props. 2.4 and 2.6) that

meas ({x∈Rⁿ, A^∗(e₁(Du_jk))(x)≥τ_k/2})≤C(n, p) (τ_k)^p

Rⁿ|e1(Du_jk)(y)|^pdy

=C(n, p) (4Rk)^p

Ω(4R)^pdy≤C(n, p)

k^p meas(Ω).

Since e₂(Du_jk) is bounded inL¹(Rⁿ) and is also supported in Ω, we apply the weak-(1,1) estimate to opera- torA^∗:

meas ({x∈Rⁿ, A^∗(e₂(Du_jk))(x)≥τ_k/2})≤ C(n) τ_k

Rⁿ|e₂(Du_jk(y))|dy

= C(n) 4Rk

{x∈Ω,|e(Du_jk(y))|≥4R}|e(Du_jk(y))|dy≤ C(n)

4Rka_jk≤ C(n) 4Rke^k+1· Thus

meas((H_jk)^c)≤ C(n, p)

k^p + C(n)

Rke^(k+1) →0. (3.13)

Consequently,

J_p(v_k,(H_jk)^c)≤C(n, p)R^p(1 +k)^pmeas((H_jk)^c)≤C(n, p)

R^p 1 +k

k _p

+k^p−1 e^k+1

≤C(n, p, R),

which follows from the simple facts that (k+ 1)/k≤2 fork= 1,2, . . . andk^p−1/e^k+1 →0 ask→ ∞. Finally, we sum up these inequalities to obtain

J_p(v_k,Rⁿ) =J_p(v_k,Ω∩W_jk) +J_p(v_k,(W_jk)^c)≤C(n, p, R) :=C₀.

The proof is then finished.

Note from (3.4) and (3.13), we have meas(W_j^c_k)→0 ask→ ∞. Lemma 3.5. Let v_k be defined as above, then

k→∞lim

Ω|e(Dvk(x))−e(Du_jk(x))|dx= 0.

Proof of Lemma 3.5. Sincee(Dv_k(x)) =e(Du_jk(x)) almost everywhere onW_jk defined by (3.8), we have

Ω|e(Dv_k(x))−e(Du_jk(x))|dx≤

(W_jk)^c|e(Dv_k(x))−e(Du_jk(x))|dx

≤J₁(v_k,(W_jk)^c) +J₁(u_jk,Ω\W_jk).

Since meas((W_jk)^c)→0 asj → ∞, andJ_p(v_k,(W_jk)^c) is bounded, we have, from H¨older’s inequality that 0≤J₁(v_k,(W_jk)^c)≤(J_p(v_k,(W_jk)^c)^1/p(meas ((W_jk)^c))^p−1p →0,

as k→ ∞.

By Lemma 3.1, |e(Du_jk(x))| is equi-integrable on Ω. We see thatJ₁(u_jk,Ω\W_jk) → 0 as k → ∞. The

conclusion follows.

Remark. Recently, I. Fonseca and the referee both explained to me that Theorem 1 may also be proved by using the general theory ofA-quasiconvex functions, in particular, the work by Fonseca and M¨uller ([15], Cor. 2.18), where a second order operatorAwas proposed to treat the linear elastic strain ([15], Ex. 3.10(e)). In fact, for the orthogonal complementE^⊥ of a general subspaceE⊂M^N×nwithout rank-one matrices, ifK⊂E^⊥is compact and dist(P_E⊥(Du_j), K) → 0 in L¹(Ω), is seems a more promising approach by using the A-quasiconvexity method than the one used here. The only problem is to work out algebraically the operator A (see [15] for details).

Proof of Corollary 1. By Korn’s inequality (Prop. 2.8), we have

Ω|Dv_k(x)−A_vk|^pdx≤C(n, p)

Ω|e(Dv_k(x))|^pdx. (3.14) Since

w_k(x) =v_k(x)− 1 meas(Ω)

Ωv_k(y)dy−A_vk(x−x_k), where x_k ∈ Rⁿ is such that

ΩA_vk(x−x_k)dx= 0, we see from Poincar´e’s inequality and (3.14), we see that (w_k) is bounded in W^1,p(Ω,Rⁿ). Also because e(Dw_k(x)) = e(Dv_k(x)) in Ω, the conclusion follows from

Lemma 3.5.

4. Proof of Theorems 2–4 and examples

Proof of Theorem 3. Let us consider the quasiconvex function on linear strainsQ_edist(A, K_α),A∈M_sⁿ. We first prove that K := {A ∈ M_sⁿ, Q_edist(A, K_α) = 0} =K_α. Obviously we have K_α ⊂K. We only need to show thatK⊂K_α. LetA₀∈K. We have from Proposition 2.1 for quasiconvex envelope on linear strains that there is a sequence (φ_j)⊂C₀^∞(D,Rⁿ) such that

j→∞lim

D

dist(A₀+e(Dφ_j(x)), K_α)dx= 0,

where D ⊂ Rⁿ is the unit cube. Let K_α,A0 = {Y −A₀, Y ∈ K_α}, then K_α,A0 is still compact and lim_j→∞

Ddist(e(Dφ_j(x)), K_α,A0)dx = 0. We then have, from Lemma 3.1 that |e(Dφ_j)| is equi-integrable onD, hence up to a subsequence (still denoted by the same subscripts)e(Dφ_j) converges weakly inL¹(D). For each fixedj, we extendφ_j to be defined inRⁿ as a periodic function and then let

u_j(x) = 1 jφ_j(jx)