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

We give a new approach to study the lower semicontinuity properties of nonautonomous 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.


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 Carathéodory function defined in Ω × R m × R mn , quasiconvex with respect to z, in Morrey's sense, that is, for every (x 0 ,s 0 ,z 0 ) ∈ Ω × R m × R mn and ϕ ∈ C ∞ 0 (Ω,R m ) there holds f x 0 ,s 0 ,z 0 |Ω| ≤ Ω f x 0 ,s 0 ,z 0 + Dϕ(y) dy. (1. 2) The function E : Ω × R m → R is a positive Carathéodory's and Φ is an N-function.
A convex function Φ : [0,+∞[ → [0,+∞[ is called N-function if it satisfies the following conditions: Φ(0) = 0, Φ(t) > 0 for t > 0, and lim t→0 Φ(t) t = 0, lim 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 (1.4) Acerbi and Fusco [5]and Marcellini [11] give a well-known weak lower semicontinuity theorem, when f is quasiconvex in Morrey's sense and satisfies the standard growth.
Theorem 1.1.Let Ω be an open set in R m .Assume that f = f (x,s,z) is a real Carathéodory function defined in Ω × R m × R mn , quasiconvex with respect to z in 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, and a ∈ L 1 loc (Ω).Then the functional is sequentially lower semicontinuous in the weak topology of W 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 energy.Consequently, this will entail a full characterization of the Young measure generated by the sequence under consideration.
In [7], there is a new proof of Theorem 1.1 by 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. ) where {ν x } x∈Ω is the Young measure generated by a subsequence of {Du j } j∈N .
In the proof of the Jensen's inequality for Young measures, an approximation theorem is fundamental which is an improvement of the result obtained by Acerbi and Fusco in [1] in the framework of Orlicz-Sobolev spaces.
The Jensen inequality is the main tool of the following theorem.
Then we can find a subsequence {u l } l of {u j } j∈N with the following properties: (1) Since f u (x,λ) = f (x,u(x),λ) satisfies the assumptions of Theorem 1.2, we have for a.e.x.Now it suffices to note that (1.12) The sequence f (x,u l (x),Du l (x)) is weakly convergent in L 1 (Ω \ E k ), then by dominate convergence the last inequality holds for Theorem 1.2.Now by the fact that |E k | → 0, we have that in Ω, (1.9) is true, then liminf j→+∞ I u j ≥ I(u). (1.14)

Notations and preliminaries
We denote by We recall some definitions and known properties of N-functions and Orlicz spaces (see [9,14]).
In the sequel, we will often use the following convexity inequality: for every s, t ∈ R and λ > 1, Now we will consider a special class of N-functions.

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Definition 2.1.An N-function Φ satisfies the ∆ 2 condition, that is Φ ∈ ∆ 2 , if there exist r > 1 and t 0 ≥ 0 such that for every t ≥ t 0 and λ > 1 there holds there exist r > 1 and t 1 ≥ 0 such that for every t ≥ t 1 and k > 1 there holds For further properties of N-functions of classes ∆ 2 and ∇ 2 , see [2,9,10,14].Let Ω be an open bounded set of R n , the Orlicz class K Φ (Ω,R m ) is the set of all equivalence classes modulo equality a.e. in Ω of measurable functions u : (2.4) The Orlicz space L Φ (Ω,R m ) is defined to be the linear hull of ) is defined to be the set of all functions in L Φ (Ω,R m ) whose first-order distributional derivatives are in L Φ (Ω,R m ).In the sequel, for a fixed λ > 0 we will consider the convex functional set (2.5)

An approximation theorem
In this section, we give an approximation theorem for functions in W 1,Φ,1 ; we will use the properties of the maximal function, some of which are related with the properties of N-functions.For details see [15].
and if We state particular properties for the maximal function (see [8,12]).
Theorem 3.2.Let Φ ∈ ∆ 2 be an N-function and f a given positive function in Then if there exists a constant a > 1 such that The maximal function M permits to control the difference quotient of u ∈ W 1,1 loc (R n ) outside a set of small measure.Definition 3.3.Set, for any u ∈ W 1,1 loc (R n ) and for any λ > 0, We now give other properties which relate the maximal function and the Nfunction.
Lemma 3.5.Let Φ be an N-function, then for every f ∈ L 1 (R n ) and for every constant λ > 0, (3.7) Proof.By the Jensen inequality applied to the convex function Φ, we obtain for the monotonicity of Φ there is (3.9) The last inequality is a property of the maximal operator M [15].
In the sequel, we will use the following approximation result (for the proof see [6]).

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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 theorem.
, and let σ be as in Lemma 3.6.
In this framework we prove that if f , then a subsequence, which we will denote by f k , exists, such that In fact, by the property applying Lemma 3.5 in (3.11) and the modular convergence, we have Let E ⊂ R n be the set with |E| = 0 and We define another sequence as (3.15) By the modular convergence and since u ∈ W 1,Φ,1 (Ω,R m ), we have For h > 0, define where c 1 is the constant of Lemma 3.4.
Since M u k → M u a.e. on Ω, then a constant k 0 exists such that, for every k > k 0 , and for Lemma 3.4 and we can conclude that the function u is h-Lipschitz continuous in Ω h,u .Furthermore, note that and then The first term becomes for the second term, we compute (3.24)By (3.23) and (3.24), we have As is well known, there exists an h-Lipschitzian function u h : Ω → R m with u h ≡ u a.e. in Ω h,u and sup Ω |u h | = sup Ωh,u |u| for every h > 0.Moreover, {x ∈ Ω : u h = u} = Ω \ (Ω h,u \ E) and |E| = 0. Then Since Ω h,u is a measurable open bounded set, then for every h > 0 there exists a closed set Then Du ≡ Du h a.e. in F h ; hence In fact, By assumption, Φ ∈ ∇ 2 and Φ(|Du|) ∈ L 1 (Ω,R m ) and by Theorem 3.2, M(Φ(|Du|)) ∈ L 1 (Ω) holds, so we have and we obtain (3.29).Finally for the monotonicity of Φ and the property of u h , we have which completes the proof.

Young measures and Jensen inequality
In this section, we give the proof of Theorem 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 measures ν = {ν x } x∈Ω associated with a sequence of functions f j : Ω ⊂ R n → R m , such that supp(ν x ) ⊂ R m , depending measurably on x ∈ Ω in the sense that for any continuous φ : R m → R, the function of where g : [0,+∞) → [0,+∞] is a continuous nondecreasing function and lim x→+∞ g(x) = +∞, then by Young existence theorem, there exist a subsequence, not relabeled, and a family of probability measures {ν x } x∈Ω (the associated Young measure) depending measurably on x with the property that whenever the sequence {ψ(x, u j (x))} j∈N is weakly convergent in L 1 (Ω) for any Carathéodory function ψ(x,λ) : Finally if we have that u j = (w j ,v j ) : Ω → R m × R k generates the Young measure {µ x } x∈Ω , w j → w in measure and that the sequence {v j } j∈N generates the Young measure {ν x } x∈Ω , then for almost every x ∈ Ω we have µ x = δ w(x) ⊗ ν x , which means that for any Before giving the proof of Jensen inequality of Theorem 1.2 we need the following proposition.Proposition 4.1.Let λ ∈ R mn and let f : R mn → R be a continuous function such that ) Proof.Take ε > 0; according to the Biting lemma (see [13]), we can find a set By the Young existence theorem, f ,ν x is the weak limit of f (Du j ); hence by Theorem 1.3 for all j ∈ N, For the second term of (4.6), sup Passing to the limit for k in (4.7) and (4.9), we have f ,ν k x → f ,ν x in L 1 (Ω \ E) as k → +∞ and the theorem is proved.
Finally we get the Jensen's inequality.
Proof of Theorem 1.2.We can assume that Ω is a ball of R n .
Step 1. Suppose that f = f (λ) is continuous and u j ∈ W 1,∞ (Ω,R m ) and take x ∈ Ω and r > 0 such that Q(x,r) ⊂ Ω.Let σ be a constant with 0 < σ < r and φ σ ∈ C ∞ 0 (Q(x,r)), where φ σ ≡ 1 on Q(x,r − σ).Applying standard arguments, the function w j σ = φ σ • (u j − u) can be substituted in the definition of quasiconvexity; hence we have for all A ∈ R mn . Let Since {Dw j σ } is uniformly bounded on Ω, the sequence { f (A + Dw j σ )} j∈N is relatively compact in L 1 (Ω).We define Then, by the definition of ḡ, the following holds: .17) Hence by (4.17), we have where ν y is supported on a bounded set.Applying the Lebesgue convergence theorem and taking the limit on σ, we have

.19)
By the theorem about the Lebesgue's points for every A ∈ R mn , a set Ω(A) ⊂ Ω exists such that |Ω \ Ω(A)| = 0 and for all x ∈ Ω(A),

.20)
Furthermore, we can suppose that sup x∈Ω(A) |Du(x)| < ∞.Let {A j } be a dense and countable subset of R mn .

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Consider now the set Ω 1 = j Ω(A j ) of full measure in Ω Hence for every x ∈ Ω 1 , (4.20) holds with A = A j .Let x ∈ Ω 1 , then by density, there exists a subsequence A jk which converges to Du(x) ∈ R mn as k → +∞.Finally, it is sufficient to note that f (A jk ) → f (Du(x)) by continuity of f , and Furthermore, by the equality Du(x) = R mn λdν x (λ) = λ,ν x and by (4.20) we have and we can conclude that f
{Du l } l generates the Young measure {ν x } x∈Ω , (3) there exists a family {E k } k∈N of sets such that |E k | → 0 as k → +∞ and { •, • the Euclidean scalar product in R n and by | • | the usual Euclidean norm.Throughout the paper, Ω denotes an open and bounded subset of R n with Lipschitz boundary.We denote by | | the Lebesgue measure on R n and the notation a.e.stands for almost everywhere with respect to Lebesgue measure.We use standard notations for spaces of classically differentiable functions, Lebesgue and Sobolev spaces.