A NEW PROOF OF SEMICONTINUITY BY YOUNG MEASURES AND AN APPROXIMATION THEOREM IN ORLICZ-SOBOLEV SPACES

A NEW PROOF OF SEMICONTINUITY BY YOUNG MEASURES AND AN APPROXIMATION THEOREM IN ORLICZ-SOBOLEV SPACES

BARBARA BIANCONI Received 6 December 2002

We give a new approach to study the lower semicontinuity properties of nonau- tonomous variational integrals whose energy densities satisfy general growth conditions. We apply the theory of Young measures and properties of Orlicz- Sobolev spaces to prove semicontinuity result.

1. Introduction

In the last years there has been a particular interest in the research of minimizers of nonautonomous variational integrals whose energy densities satisfy general growth conditions such as

0≤f(x, s, z)≤E(x, s)1 +Φ|z|

, (1.1)

where f = f(x, s, z) is a real Carath´eodory function defined inΩ×R^m×R^mn, quasiconvex with respect toz, in Morrey’s sense, that is, for every (x0, s0, z0)∈ Ω×R^m×R^mnandϕ∈C0^∞(Ω,R^m) there holds

fx0, s0, z0

|Ω| ≤

Ωfx0, s0, z0+Dϕ(y)d y. (1.2) The functionE:Ω×R^m→Ris a positive Carath´eodory’s andΦis anN-func- tion.

A convex functionΦ: [0,+∞[→[0,+∞[ is calledN-functionif it satisfies the following conditions:Φ(0)=0,Φ(t)>0 fort >0, and

limt→0

Φ(t)

t ⁼0, lim

t→+∞

Φ(t)

t ⁼+∞. (1.3)

WhenΦ(z)=z^p, we say that f verifiesstandardgrowth conditions.

Copyright©2003 Hindawi Publishing Corporation Abstract and Applied Analysis 2003:15 (2003) 881–898 2000 Mathematics Subject Classification: 49J45 URL:http://dx.doi.org/10.1155/S1085337503303045

The study of nonautonomous variational integrals is relevant for studying the applications in the theory of elastic and magnetostatic material behaviors. Often a starting point is the necessary and sufficient conditions ensuring sequential weak lower semicontinuity of the functional

F(u)=

Ωfx, u(x), Du(x)dx. (1.4) Acerbi and Fusco [5]and Marcellini [11] give a well-known weak lower semi- continuity theorem, when f is quasiconvex in Morrey’s sense and satisfies the standard growth.

Theorem1.1. LetΩbe an open set inR^m. Assume that f = f(x, s, z)is a real Carath´eodory function defined inΩ×R^m×R^mn, quasiconvex with respect tozin Morrey’s sense, and such that

0≤f(x, s, z)≤a(x) +c|s|^p+|z|^p

for a.e.x∈Ω,∀s∈R^m,∀z∈R^mn, (1.5) where c is a positive constant,p≥1, anda∈L¹_loc(Ω).

Then the functional

u_∈W^1,pΩ;R^m−→

Ωfx, u(x), Du(x)dx (1.6) is sequentially lower semicontinuous in the weak topology ofW^1,p(Ω;R^m).

In [4], the result has been generalized by Bianconi et al. for general growth (1.1) and the lower semicontinuity in the weak^∗topology of the Orlicz-Sobolev spaces is proved.

In some physical problems, there may be situations where we need to identify lim_n→∞F(u_n) for an oscillatory sequence{u_n}which does not minimize the en- ergy. Consequently, this will entail a full characterization of the Young measure generated by the sequence under consideration.

In [7], there is a new proof ofTheorem 1.1by using Young measures. In this setting the semicontinuity is a direct consequence of the Jensen inequality.

In this paper, we give a new proof of the lower semicontinuity for quasiconvex integrals satisfying (1.1) in the framework of Young measures.

The first step is the Jensen-type inequality for Young measures in Orlicz- Sobolev spaces.

Theorem1.2. LetΩ⊂Rⁿbe a bounded set,u^j∈W^1,Φ,1(Ω,R^m), andΦ∈∆2∩

∇2. Suppose thatu^j→uinL¹_locnorm andlim infj

ΩΦ(|Du^j(x)|)dx <+∞. Let f :Ω×R^mn→[0,+∞]verifying that

(a) f(x, λ)is a Carath´eodory function,

(b)a measurable functionE:Ω→Rexists such that for almost everyx∈Ω and for allλ∈R^mn, f(x, λ)≤E(x)(1 +Φ(|λ|))holds,

(c)the functionλ→f(x, λ)is quasiconvex for almost everyx∈Ω. Then

f

x,

R^mnλ dνx(λ)

≤

R^mn f(x, λ)dνx(λ), (1.7) Du(x)=

R^mnλ dνx(λ), (1.8)

where{νx}x∈Ωis the Young measure generated by a subsequence of{Du^j}j∈N. In the proof of the Jensen’s inequality for Young measures, an approxima- tion theorem is fundamental which is an improvement of the result obtained by Acerbi and Fusco in [1] in the framework of Orlicz-Sobolev spaces.

Theorem 1.3. Let Ω⊆Rⁿ be the unit ball and u∈W^1,Φ,1(Ω,R^m)with Φ a genericN-function, then for every constanth >0, there exist a functionu_h∈Lip(Ω, R^m)and a closed setF_h⊂Ωsuch that

(i)Φ(∇u_h)_L^∞_(Ω,R^m)≤Φ(h), (ii)∇u= ∇uha.e. inFh, (iii) limh→+_∞|Ω\Fh| =0,

(iv)ifΦ∈∆2∩ ∇2, thenlim_h→+∞Φ(h)|Ω\F_h| =0.

The Jensen inequality is the main tool of the following theorem.

Theorem1.4. LetΩ⊆Rⁿbe a bounded set and let f :Ω×R^m×R^mn→[0,+∞] with the following properties:

(1) f(x, s, λ)is a Carath´eodory function,

(2)a Carath´eodory functionE(·,·)exists such that, for a.e.xand for almost (s, λ), f(x, s, λ)≤E(x, s)(1 +Φ(|λ|)), whereΦis anN-function withΦ∈

∆2∩ ∇2,

(3)for a.e.xand for alls, the mappingλ→f(x, s, λ)is quasiconvex.

Then for every{u^j}j∈N,u^j∈W^1,Φ,1(Ω,R^m)such thatu^j→uinL¹_loc(Ω,R^m)and lim infj

ΩΦ(|Du^j(x)|)dx <+∞,

I(u)≤lim inf

j Iu^j. (1.9)

Proof. Letα=lim inf_jI(u^j). Ifα=+∞, the assertion is satisfied. Suppose that α <+∞. In this case the sequence{f(x, u^j(x), Du^j(x))}j∈Nis bounded inL¹(Ω).

Then we can find a subsequence{u^l}lof{u^j}j∈Nwith the following properties:

(1)I(u^l)→αasl→+∞,

(2){Du^l}lgenerates the Young measure{νx}x∈Ω,

(3) there exists a family{Ek}k∈Nof sets such that|Ek| →0 ask→+∞and {f(x, u^l(x), Du^l(x))}l is weakly convergent in L¹(Ω\E_k) to _Rmn f(x, u(x), λ)dνx(λ).

Since fu(x, λ)=f(x, u(x), λ) satisfies the assumptions ofTheorem 1.2, we have

R^mnf_u(x, λ)dνx(λ)≥f_ux, Du(x), (1.10) hence

R^mnfx, u(x), λdνx(λ)≥ fx, u(x), Du(x) (1.11) for a.e.x. Now it suffices to note that

α=lim inf

j→+_∞ Iu^j= lim

l→+_∞

Ωfx, u^l(x), Du^l(x)dx

≥ lim

l→+_∞

Ω\Ek

fx, u^l(x), Du^l(x)dx.

(1.12)

The sequence f(x, u^l(x), Du^l(x)) is weakly convergent in L¹(Ω\E_k), then by dominate convergence

l→lim+_∞

Ω\Ek

fx, u^l(x), Du^l(x)dx=

Ω\Ek

R^mnfx, u(x), λdνx(λ)dx

≥

Ω\Ek

fx, u(x), Du(x)dx

=I(u)_|Ω\Ek,

(1.13)

the last inequality holds forTheorem 1.2. Now by the fact that|Ek| →0, we have that inΩ, (1.9) is true, then

lim inf

j→+∞ Iu^j≥I(u). (1.14)

2. Notations and preliminaries

We denote by·,·the Euclidean scalar product inRⁿand by| · |the usual Eu- clidean norm. Throughout the paper,Ωdenotes an open and bounded subset of Rⁿwith Lipschitz boundary. We denote by| |the Lebesgue measure onRⁿand the notation a.e. stands for almost everywhere with respect to Lebesgue mea- sure. We use standard notations for spaces of classically differentiable functions, Lebesgue and Sobolev spaces.

We recall some definitions and known properties ofN-functions and Orlicz spaces (see [9,14]).

In the sequel, we will often use the following convexity inequality: for everys, t∈Randλ >1,

Φ(s+t)≤1

λΦ(λs) +1−1 λ

Φ λ λ−1t

. (2.1)

Now we will consider a special class ofN-functions.

Definition 2.1. An N-function Φsatisfies the ∆2 condition, that isΦ∈∆2, if there existr >1 andt0≥0 such that for everyt≥t0andλ >1 there holds

Φ(λt)≤λ^rΦ(t). (2.2) Definition 2.2. AnN-functionΦsatisfies the∇2condition, that isΦ∈ ∇2, if there existr >1 andt1≥0 such that for everyt≥t1andk >1 there holds

Φ(kt)≥k^rΦ(t). (2.3)

For further properties ofN-functions of classes∆2and∇2, see [2,9,10,14].

LetΩbe an open bounded set ofRⁿ, theOrlicz classK^Φ(Ω,R^m) is the set of all equivalence classes modulo equality a.e. inΩof measurable functionsu:Ω→ R^msatisfying

ΩΦ|u|

dx <+∞. (2.4)

TheOrlicz spaceL^Φ(Ω,R^m) is defined to be the linear hull ofK^Φ(Ω,R^m).

TheOrlicz-Sobolev spaceW¹L^Φ(Ω,R^m) is defined to be the set of all functions inL^Φ(Ω,R^m) whose first-order distributional derivatives are inL^Φ(Ω,R^m). In the sequel, for a fixedλ >0 we will consider the convex functional set

W^1,Φ,λΩ,R^m= u∈W^1,1Ω,R^m:

ΩΦλ|Du|

dx <+∞

. (2.5)

3. An approximation theorem

In this section, we give an approximation theorem for functions inW^1,Φ,1; we will use the properties of the maximal function, some of which are related with the properties ofN-functions. For details see [15].

Definition 3.1. For every f ∈L¹_loc(Rⁿ), set M f(x)=sup

r>0

B(x, r1 )

B(x,r)

f(y)d y, (3.1)

and if f ∈W_loc^1,1(Rⁿ), setMf(x)=M(|∇f|), that is, Mf(x)=

n i=1

M f_xi(x). (3.2)

We state particular properties for the maximal function (see [8,12]).

Theorem3.2. LetΦ∈∆2 be anN-function and f a given positive function in L¹_loc(Rⁿ).

Then if there exists a constanta >1such that Φ(t)< 1

2aΦ(at), t≥0, (3.3)

then

RⁿΦM f(x)dx≤c

RⁿΦf(x)dx. (3.4) The maximal functionMpermits to control the difference quotient ofu∈ W_loc^1,1(Rⁿ) outside a set of small measure.

Definition 3.3. Set, for anyu∈W_loc^1,1(Rⁿ) and for anyλ >0, Hλ,u=

x∈Rⁿ:Mu(x)< λ. (3.5) Lemma3.4. There exists a constantc1=c1(n)such that, for everyu∈C0^∞(Rⁿ,R^m),

u(x)−u(y)

|x−y| ≤c1λ ∀x, y∈Hλ,u. (3.6) We now give other properties which relate the maximal function and theN- function.

Lemma 3.5. LetΦbe an N-function, then for every f ∈L¹(Rⁿ)and for every constantλ >0,

Φ(λ)x∈Rⁿ:M f(x)≥λ≤

RⁿΦf(x)dx. (3.7) Proof. By the Jensen inequality applied to the convex functionΦ, we obtain

ΦM f(x)≤MΦf(x); (3.8) for the monotonicity ofΦthere is

Φ(λ)x∈Rⁿ:M f(x)≥λ=Φ(λ)x∈Rⁿ:ΦM f(x)≥φ(λ)

≤Φ(λ)x_∈Rⁿ:MΦf(x)_≥Φ(λ)

≤

RⁿΦf(x)dx.

(3.9)

The last inequality is a property of the maximal operatorM[15].

In the sequel, we will use the following approximation result (for the proof see [6]).

Lemma3.6. LetΩbe a regular open set ofRⁿ; ifu∈W^1,Φ,1(Ω,R^m), there exist a constantσ >0and a sequenceuk∈C^∞0(Ω,R^m)which converges modularly tou:

uk σ

−−→mod u, that is,

klim→+∞

ΩΦDu(x)−Du_k(x) σ

dx=0. (3.10)

Remark 3.7. In [6] the last result was proved only in the scalar case, but it is easy to show that it holds in general.

Now we have all the necessary ingredients to prove the approximation theo- rem.

Proof ofThereom 1.3. Letu∈W^1,Φ,1(Ω,R^m),uk∈C^∞₀(Ω,R^m), and letσbe as in Lemma 3.6.

In this framework we prove that if f_k∈C^∞₀(Rⁿ,R^m) and f ∈W^1,Φ,1(Rⁿ,R^m) with f_k−−→^σ

mod f, then a subsequence, which we will denote by f_k, exists, such that Mfk→Mf in measure.

In fact, by the property|Mg1(x)−Mg2(x)| ≤M(g1−g2)(x), forα >0, there is

x∈Rⁿ:Mf(x)−Mfk(x)≥σα

≤

x∈Rⁿ:MD f(x)−D f_k(x) σ

≥α;

(3.11)

applyingLemma 3.5in (3.11) and the modular convergence, we have x∈Rⁿ:Mf(x)−Mf_k(x)≥σα

≤ 1 Φ(α)

RⁿΦ D f(x)−D f_k(x) σ

dx−−−−→^k→+∞ 0;

(3.12)

therefore,Mfk→Mf in measure, then a subsequenceMfkj ofMfk, which we will denote byMfk, exists such thatMfk→Mf a.e. inRⁿ.

LetE⊂Rⁿbe the set with|E| =0 and Rⁿ\E=

x∈Rⁿ:fk(x)−→ f(x), Mfk(x)−→Mf(x). (3.13) We define another sequence as

vk(x) := 1

1 +σuk(x) ∀k∈N. (3.14)

Thenvk∈C₀^∞(Ω,R^m) for allk >0. Letv^∗_k be the natural extension ofvkonRⁿ, still denoted withvk, then compute

RⁿΦDvk(x)dx≤

Ω

σ

1 +σΦDuk(x)−Du(x)dx +

Ω

1

1 +σΦDu(x)dx

= σ 1 +σ

ΩΦ Du_k(x)−Du(x) σ

dx

+ 1

1 +σ

ΩΦDu(x)dx.

(3.15)

By the modular convergence and sinceu∈W^1,Φ,1(Ω,R^m), we have

RⁿΦDv_k(x)dx <Γ<+∞; (3.16) therefore,v_k∈W^1,Φ,1(Rⁿ,R^m).

Forh >0, define

Ωh,u= x∈Ω:Mu(x)< h c1

, (3.17)

wherec1is the constant ofLemma 3.4.

SinceMuk→Mua.e. onΩ, then a constantk0exists such that, for every k > k0,

Mu_k(x)< h c1

a.e.x∈Ωh,u, (3.18)

and forLemma 3.4

u_k(x)−u_k(y)

|x−y| ≤c1h

c1 =h a.e.x, y∈Ωh,u. (3.19) Ask→+∞we obtain

u(x)−u(y)

|x−y| ≤h a.e.x, y∈Ωh,u, (3.20) and we can conclude that the functionuish-Lipschitz continuous inΩh,u.

Furthermore, note that

Ω\Ωh,u= x∈Ω:Mu(x)≥ h c1

= x∈Ω:Mu(x)−Mu_k(x) +Mu_k(x)≥ h c1

⊂ x∈Ω:Mu(x)−Muk(x)+Muk(x)≥ h c1

⊂ x∈Ω:Mu(x)−Muk(x)≥ h 2c1

∪ x∈Ω:Mu_k(x)≥ h 2c1

,

(3.21) and then

Ω\Ωh,u≤

x∈Ω:Mu(x)−Muk(x)≥ h 2c1

+ x∈Ω:Muk(x)≥ h

2c1

.

(3.22)

The first term becomes

x∈Ω:Mu(x)−Muk(x)≥ h 2c1

=

x∈Ω:Mu(x)−Muk(x)

σ ^≥

h 2c1σ

≤ 1

Φh/2c1σ

ΩΦDuk(x)−Du(x) σ

dx;

(3.23)

for the second term, we compute x∈Ω:Muk(x)≥ h

2c1

=

x∈Ω:Mvk(x)≥ h 2c1(1 +σ)

≤

x∈Rⁿ:MDvk(x)≥ h 2c1(1 +σ)

≤ 1

Φh/2c1(1 +σ)

RⁿΦDv_k(x)dx

≤ Γ

Φh/2c1(1 +σ).

(3.24) By (3.23) and (3.24), we have

Ω\Ωh,u≤ 1 Φh/2c1σ

RⁿΦDu_k(x)−Du(x) σ

dx

+ Γ

Φh/2c1(1 +σ).

(3.25)

As is well known, there exists anh-Lipschitzian functionuh:Ω→R^mwithuh≡ ua.e. inΩh,uand sup_Ω|uh| =sup_Ωh,u|u|for everyh >0.

Moreover,{x∈Ω:uh=u} =Ω\(Ωh,u\E) and|E| =0.

Then

x∈Ω:uh=u=Ω\Ωh,u. (3.26) SinceΩh,uis a measurable open bounded set, then for everyh >0 there exists a closed setF_h⊂Ωh,usuch that|Ωh,u\F_h| ≤1/Φ²(h).

ThenDu≡Duha.e. inFh; hence

Ω\Fh=Ω\Ωh,u+Ωh,u\Fh≤Ω\Ωh,u+ 1

Φ²(h). (3.27) By (3.25),

hlim→+∞Ω\Fh=0 (3.28) andTheorem 1.2(c) is proved.

WhenΦ∈∆2∩ ∇2, we will prove

hlim→+∞Φ(h)Ω\F_h=0. (3.29) In fact,

Φ(h)Ω\Fh=Φ(h)Ω\Ωh,u+Φ(h)Ωh,u\Fh

≤ 2

Φ(h)+Φ(h)Ω\Ωh,u. (3.30) SinceΦ∈∆2, we can compute

Φ(h) x∈Ω:Mu(x)≥ h c1

≤cΦh c1

x∈Ω:Mu(x)≥ h c1

≤cΦh c1

x∈Ω:MΦDu(x)≥Φh c1

≤c

{x∈Ω:M(Φ(|Du|))(x)≥Φ(h/c1)}MΦ(|Du|)(x)dx.

(3.31)

By assumption, Φ∈ ∇2 and Φ(|Du|)∈L¹(Ω,R^m) and by Theorem 3.2, M(Φ(|Du|))∈L¹(Ω) holds, so we have

Φ(h) x∈Ω:Mu(x)≥ h c1

−−−−→^h→+∞ 0 (3.32)

and we obtain (3.29). Finally for the monotonicity ofΦand the property ofuh, we have

Duh≤h=⇒ΦDuh≤Φ(h)=⇒ΦDuh

L^∞(Ω,R^m)≤Φ(h), (3.33)

which completes the proof.

4. Young measures and Jensen inequality

In this section, we give the proof ofTheorem 1.2, based on arguments of the theory of Young measures. Hence we recall the most important properties; for the related proofs and particular results, we refer to [3,7,13].

A Young measure is a family of probability measuresν= {νx}x∈Ωassociated with a sequence of functions f_j:Ω⊂Rⁿ→R^m, such that supp(νx)⊂R^m, de- pending measurably onx∈Ωin the sense that for any continuousφ:R^m→R, the function ofx

φ(x)¯ =

R^mφ(λ)dνx(λ)= φ,νx

(4.1)

is measurable.

If{u_j}j∈Nis a sequence of measurable functionsu_j:Ω→R^msuch that sup

j

Ωgujdx <+∞, (4.2)

where g : [0,+∞)→[0,+∞] is a continuous nondecreasing function and limx→+_∞g(x)=+∞, then by Young existence theorem, there exist a subsequence, not relabeled, and a family of probability measures {νx}x∈Ω (the associated Young measure) depending measurably onxwith the property that whenever the sequence{ψ(x, uj(x))}j∈Nis weakly convergent inL¹(Ω) for any Carath´eo- dory function ψ(x, λ) :Ω×R^m→_R¯, the weak limit is the function ¯ψ(x)=

R^mψ(x, λ)dνx(λ).

Finally if we have thatu^j=(w^j, v^j) :Ω→R^m×R^kgenerates the Young mea- sure{µ_x}x∈Ω,w^j→w in measure and that the sequence{v^j}j∈Ngenerates the Young measure{νx}x∈Ω, then for almost everyx∈Ωwe haveµ_x=δ_w(x)⊗νx, which means that for any f ∈C0(R^m×R^k) and almost everyx∈Ω,

R^m×R^k f(s, λ)dµ_x(s, λ)=

R^k fw(x), λdνx(λ). (4.3) Before giving the proof of Jensen inequality ofTheorem 1.2we need the fol- lowing proposition.

Proposition4.1. Letλ∈R^mnand let f :R^mn→Rbe a continuous function such that

f(λ)≤c1 +Φ|λ|

(4.4) withΦ∈∆2∩ ∇2.

For{u^j}j∈NinW^1,φ,1(Ω,R^m)such that sup

j

ΩΦDuj(x)dx≤M, (4.5)

letu_k^j_∈Lip(Ω,R^m)be the Lipschitz function sequence ofTheorem 1.3. Let{νx}x∈Ω

and {ν^kx}x∈Ω be the Young measures generated by the sequences{Du^j}j∈N and {Du_k^j}j∈N, respectively. Then for all ε >0 there exists a subset E⊂Ωsuch that

|E|< εandf ,ν^kx → f ,νxinL¹(Ω\E)ask→+∞.

Proof. Takeε >0; according to the Biting lemma (see [13]), we can find a set E⊂Ωsuch that|E|< εand f(Du^j) f¯inL¹(Ω\E).

By the Young existence theorem,f ,νx is the weak limit of f(Du^j); hence f¯= f ,νxa.e. inΩ. Then,

Ω\E

f ,ν^k_x−

f ,νxdx

= sup

ψL^∞≤1

Ω\Eψ(x)f ,ν^kx

− f ,νx

dx

= sup

ψL^∞≤1

lim

j→+_∞

Ω\Eψ(x)fDu_k^j−fDu^jdx

≤ sup

ψL^∞≤1

lim

j→+_∞

Ω\E

ψ(x)fDu_k^j−fDu^jdx

= sup

ψL^∞≤1

lim

j→+_∞

(Ω\E)\F_k^j

ψ(x)fDu_k^j−fDu^jdx + sup

ψL^∞≤1

lim

j→+∞

(Ω\E)∩F_k^j

ψ(x)fDu_k^j−fDu^jdx

≤sup

j

(Ω\E)\F_k^j

fDu_k^j−fDu^jdx

≤sup

j

(Ω\E)\F_k^j

fDu_k^jdx+ sup

j

(Ω\E)\F_k^j

fDu^jdx.

(4.6)

Since

sup

j

(Ω\E)\F_k^j

fDu_k^jdx

≤sup

j

(Ω\E)\F_k^jc1 +ΦDu_k^jdx

≤sup

j

c(Ω\E)\F_k^j+ sup

j

(Ω\E)\F_k^jcΦDu_k^jdx

≤sup

j

c(Ω\E)\F_k^j+ sup

j cΦ(k)(Ω\E)\F_k^j

(4.7) byTheorem 1.3for allj∈N,

(Ω\E)\F_k^j−−−−→^k→+∞ 0, Φ(k)(Ω\E)\F_k^j−−−−→^k→+∞ 0. (4.8) For the second term of (4.6),

sup

j

(Ω\E)\F_k^j

fDu^jdx≤sup

j

(Ω\E)\F_k^jc1 +ΦDu^j. (4.9) Passing to the limit forkin (4.7) and (4.9), we havef ,ν^kx → f ,νxinL¹(Ω\E)

ask→+∞and the theorem is proved.

Finally we get the Jensen’s inequality.

Proof ofTheorem 1.2. We can assume thatΩis a ball ofRⁿ.

For the proof of (1.8), we define the Carath´eodory functionΨ^(h,s):Ω×R^mn→ R, withΨ^(h,s)(x, λ)=λh,s, whereλh,sis the element in position (h, s) of the ma- trixλ∈R^mn. Then for what recalled about Young measures,Ψ^(h,s)(x, Du^j(x)) Ψ¯^(h,s)(x)=

R^mnΨ^(h,s)(x, λ)dνx(λ) holds, as j→+∞, inL¹(Ω), and by hypothe- sis,Ψ^(h,s)(x, Du^j(x))(Du(x))h,sas j→+∞. So by uniqueness of weak limit we have

Du(x)_h,s=

R^mnλh,sdνx(λ) for almost everyx∈Ω, (4.10) for allh=1,2, . . . , mands=1,2, . . . , n, then

Du(x)=

R^mnλ dνx(λ) for almost everyx∈Ω. (4.11) Now we divide the proof into 4 steps.

Step 1. Suppose that f = f(λ) is continuous andu^j∈W^1,∞(Ω,R^m) and take x_∈Ωandr >0 such thatQ(x, r)_⊂Ω. Letσ be a constant with 0< σ < r and φ_σ∈C^∞₀(Q(x, r)), whereφ_σ≡1 onQ(x, r−σ). Applying standard arguments, the functionwσ^j=φ_σ·(u^j−u) can be substituted in the definition of quasicon- vexity; hence we have

f(A)≤ 1 Q(x, r)

Q(x,r)fA+Dφσ·

u^j−u+φσ·

Du^j−Dud y (4.12) for allA∈R^mn.