Multiscale homogenization of integral convex functionals in Orlicz Sobolev setting

The $\Gamma $-limit of a family of functionals $u\mapsto \int_{\Omega }f\left( \frac{x}{\varepsilon },\frac{x}{\varepsilon ^{2}},D^{s}u\right) dx$ is obtained for $s=1,2$ and when the integrand $f=f\left( y,z,v\right) $ is a continous function, periodic in $y$ and $z$ and convex with respect to $v$ with nonstandard growth. The reiterated two-scale limits of second order derivative are characterized in this setting.

Recently, in [18], these results have been extended to the multiscale setting, see subsection 2.2 for precise definitions and results.
The aim of this work consists of extending the latter results, together with a Γ− convergence theorem, to higher order Sobolev-Orlicz spaces under suitable assumptions on the N -function. In details we will deal with the functional where f satisfies the following hypothesis: f : R N y × R N z × R s * → [0, +∞) is such that: (H1) f is continous or (A1) f (·, z, λ) is continous for a.e. z and every λ, (A2) f (y, ·, λ) is measurable for all (y, λ) ∈ Ω × R s * , (A3) ∂f ∂λ (y, z, λ) exists for every y ∈ Ω and for a.e. z ∈ R N z , and for every λ ∈ R s * and it satisfies (A1) and (A2); (H2) f is separately Y − periodic in y and z; (H3) f (y, z, ·) is convex for all y and almost every z ∈ R N z ; (H4) There exist two constants c1,c2 > 0 such that c1B (|λ|) ≤ f (y, z, λ) ≤ c2 (1 + B (|λ|)) for all λ ∈ R s * , for a.e. z ∈ R N y and all y ∈ Ω, where Y is a copy of the unit cube (−1/2, 1/2) N , and Ω is a bounded open subset of R N , s ∈ {1, 2}, B is an N -function satisfying, together with its conjugate function, △2 condition (see [1] and Section 2 below).
Moreover R s * coincides with R d×N if s = 1 and with Sym R N , R N d , where Sym R N , R N denotes the space of all linear symetric transformations from R N to R N .
Bearing in mind that N, m, d ∈ N, ε denotes a sequence of positive real numbers converging to 0, and denoting (as above) by Y and Z two identical copies of the cube (−1/2, 1/2) N , adopting the notations in subsection 2.1, our first main result deals with the reiteratively two-scale convergence in second order Sobolev-Orlicz spaces. Indeed we have the following result: Let Ω be a bounded open set of R N , with Lipschitz boundary. If (uε)ε is a bounded sequence in W 2 L B Ω; R d , then there exist a subsequence (not relabelled) converging weakly in W 2 L B Ω; R d to a function u, and functions U ∈ L 1 Ω; The other main result deals with the Γ− convergence of the family (Fε)ε in (1.2), thus extending, from one hand, Theorem 1.1. in [17] and, from the other, generalizing to the Orlicz-Sobolev setting [14, Theorem 1.8]: When s = 1 we will denote f s hom simply by f hom . We emphasize that the above results could be recasted in the framework of Periodic Unfolding, introduced in [10], (see also [11] for a systematic treatment) or applied to the non convex case, in the spirit of [9], and these are indeed the subjects of our future investigation.
In the next section we establish notation and recall some preliminary results, mainly adopting the symbols already used in [16], and [18], while Section 3 is devoted to establish Theorems 1.1 and 1.2.

Preliminaries
In the sequel, in order to enlighten the space variable under consideration we will adopt the notation R N x , R N y , or R N z to indicate where x, y or z belong to. On the other hand, when it will be clear from the context, we will simply write R N .
The family of open subsets in R N will be denoted by A(R N ), while the family of Borel sets is denoted by B(R N ).
For any subset D of R m , m ∈ N, by D, we denote its closure in the relative topology. Given an open set A by C b (A) we denote the space of real valued continuous and bounded functions defined in A.
For every x ∈ R N we denote by [x] its integer part, namely the vector in Z N , which has as components the integer parts of the components of x.
By L N we denote the Lebesgue measure in R N . Now we recall results of Orlicz-Sobolev spaces that will be used in the remainder of the paper.
In all what follows B and B are conjugates N −functions both satisfying the △2 condition and c refers to a generic constant that may vary from line to line. Let Ω be a bounded open set in R N . The Orlicz-space is a Banach space for the Luxemburg norm: It follows that: D (Ω) is dense in L B (Ω) , L B (Ω) is separable and reflexive, the dual of L B (Ω) is identified with L B (Ω) , and the norm on L B (Ω) is equivalent to · B,Ω . Futhermore, it is also convenient to recall that: Let s = 1 or 2, following [2] one can introduce the Orlicz-Sobolev space W s L B (Ω; R d ), consisting of those (equivalence classes of) functions u ∈ L B (Ω)) for which D s u ∈ L B (Ω; R s * ), and the derivatives are taken in the distributional sense on Ω. For s = 1, . It is immediately seen the extension to vector fields We denote by W s 0 L B (Ω) , the closure of D (Ω) in W s L B (Ω) and the semi-norm . Arguing in components, the same definitions hold for W s L B (Ω; R d ) and W s 0 L B (Ω; R d ). By W s # L B (Y ), we denote the space of functions u ∈ W s L B (Y ) such that Y u(y)dy = 0. Given a function space S defined in Y , Z or Y × Z, the subscript Sper means that the functions are periodic in Y , Z or Y × Z, as it will be clear from the context. In particular by Cper(Y ), Cper(Z) (or Cper(Y × Z) respectively), we denote the space of continuous functions in R d , which are Y or Z-periodic (continuous function in R d × R d , which are Y × Z-periodic, respectively).

Reiterated two scale convergence in first order Sobolev-Orlicz spaces.
In the sequel we present a generalization of definitions in [16,23,26] obtained in [18]. To this end, we recall that, within this section, Ω is a bounded open set with Lipschitz boundary, and we denote by L B per (Ω × Y × Z) the space of functions in v ∈ L B loc (Ω × R N × R N ) which are periodic in Y × Z and such that Ω×Y ×Z B(|v|)dxdydz < +∞. For any given ε > 0 and any function v ∈ L B (Ω; Cper(Y × Z)), we define This function is well defined as proven in [ When (2.2) happens we denote it by "uε ⇀ u0 in L B (Ω) − weakly reiteratively twoscale " and we will say that u0 is the weak reiterated two-scale limit in L B (Ω) of the sequence (uε) ε .
The above definition extends in a canonical way, arguing in components, to vector valued functions.
Moreover for the sake of exposition, the reiterated weak convergence of uε towards u0 in L B will be also denoted by the symbol uε reit−2s ⇀ u0, both in the scalar and in the vector valued setting.
The proof of the following lemma can be found in [18, Proof of Lemma 2.3].
The subsequent results, useful in the remainder of the paper, explicitly for the construction of sequences which ensure the energy convergence in Theorem 1.2, have been proven in [18, Section 2.3].
endowed with the norm: The following is a sequential compactness result on W 1 L B (Ω) , (see [16] and [18] for a proof and related results) that will be used in the sequel.
If (i) and (ii) in the above Proposition hold, we will say that u ε ⇀ u0 reiteratively weakly two-scale in W 1 L B (Ω), omitting to explicitly mention the functions u1, u2 above.

Remark 2.2.
We observe that the fields (u1, u2) in Proposition 2.4 are more regular than stated above and in [17]. Indeed by (ii), Thus, applying Poincaré-Wirtinger inequality, (see for instance [8,Theorem 4.5]) to u1 with respect to Y and for a.e. x ∈ Ω and to u2 with respect to z for a.e. x ∈ Ω and for any y ∈ Y , and then taking the integral over Ω for u1 and over Ω × Y for u2, it is easily seen that the L B norm in Ω × Y of u1 is finite, and the same holds for the L B norm in . Moreover, it is worth to observe that the same convergence holds for vector valued functions.
u0 reiteratively weakly two-scale in W 1 L B (Ω), we have: Under our sets of assumptions on Ω and B, the canonical injection W 1 L B (Ω) ֒→ L B (Ω) is compact, an so the reiterated weakly two-scale limit u0 ∈ W 1 L B (Ω) .

Γ convergence and preliminary results on integral functionals.
In the sequel we recall the definition of Γ-convergence in metric spaces. We refer to [12] for a complete treatment of the subject.
Definition 2.2. Let (X, d) be a metric space and let (Fε) ε>0 be a family of functionals defined on (X, d) . We say that a functional F : Next we recall for the readers' convenience Ioffe's lower semicontinuity theorem (see [19,Theorem 1]).
be a function Lebesgue measurale with respect to x and Borel measurable with respect to (u, v) .
The following results will be used in the sequel. We omit their proofs since they are entirely similar to their counterparts in the classical Sobolev setting (see [14,Appendix] or [6, Lemma 3.3] for similar arguments).
For every sequence (εn) converging to 0 + there exists a subsequence (εj) such that the functional F {εj} (u, D) is the Γ L B (D) limit of (Fε j (u, D)) for every D ∈ A (Ω) and

Proof of Main results
This section is devoted to the proof of our main results and it extends to the Orlicz-Sobolev setting, the arguments in [14]. Thus we do not give all the details, but we present just the proofs which involve different techniques and estimates.
We start recalling that, within this section Ω ⊂ R N is a bounded open set with Lipschitz boundary.
The proof of our first result is a consequence of the analogous theorem in [14] and the assumption (H), i.e. that the N -function B satisfies the assumption that there exists p, q > 1 such that (H) L q (Ω) ֒→ L B (Ω) ֒→ L p (Ω).
On the other hand, this assumption is satisfied by any N -function B such that △2 condition holds both for B and for its conjugate. Indeed it suffices to apply [13,Proposition 2.4] (see also [7,Proposition 3.5]) and standard rearrangements' arguments, which allow to consider any dimension N .
Proof of Theorem 1.1. We start recalling the following property which will be used in the sequel: since Ω has Lipschitz boundary, it is well known that if v ∈ L 1 (Ω) and its distributional gradient ∇v ∈ L B (Ω), then v ∈ W 1 L B (Ω). Clearly the same properties are shared by vector valued fields.
Let assume that (uε) ε is bounded in W 2 L B Ω; R D , with B satisfying △2 and (H). Hence uε W 2,p ≤ c uε W 2 L B and (uε) ε bounded in W 2 L B implies (uε) ε bounded in W 2,p where the counterparts of (i)-(iv) in the Sobolev setting are known, see [14]. Moreover since (uε) ε is bounded in W 2 L B Ω; R d we have that for every j, ∂uε ∂x j ε is bounded in On the other hand the strong convergence of uε → u0 in W 1 L B Ω; R d , and the bounds on the hessians, together with proposition 2.4, applied to (Duε)ε, entail that u 1 0 and u 2 0 = 0.
Averaging over zi, we deduce and consequently From (3.1) and p 1 0 ∈ L B Ω; W 1 # L B Y ; R d , we have for a.e x ∈ Ω, p 1 0 (x, ·) ∈ W 1 # L B Y ; R d , then Arguing as in Remark 2.2, we can actually prove that U ∈ L B Ω; On the other hand, [14, Theorem 1.7] guarantees the existence of a field A (x) ∈ L p Ω; . In a similar way, Then, the existence of a field C such that for some C ∈ L 1 Ω × Y ; R d×N , can be deduced arguing as above relying on the analogous property proven in [14, Theorem 1.10].

Proof of Theorem 1.2.
The result is achieved adopting the same strategy as in [14], i.e. by means of the followings lemmas. The first one deals with the continuity of f hom .
Thus by (H4), we deduce 1 C 1 C 2 Q B (|ξn + Dψn (y)|) dy < 1. Recalling that B is convex and B (0) = 0, it is easily seen that Q B |ξn+Dψn(y)| Thus, exploiting the definition of L B norm, the triangle inequality and the convergence of ξn to ξ, we have that From Poincaré-Wirtinger's inéquality, the fact that B satisfies △2 condition, hence is reflexive, it results that, up to a not relabelled subsequence, ψn − Q ψndy ⇀ ψ in W 1 per L B Q; R d ; thus Dψn ⇀ Dψ in L B Q; R d . In view of (A1) , (A2) and by theorem 2.5, we get that: Clearly, under the same assumptions, the above result holds for f hom , f 2 hom and f 2 hom . We are in position to introduce a localized version of our Γ-limit, i.e. we set for any sequence (εn)n of positive real numbers converging to zero, where, with an abuse of notation (cf. (1.2)) we define for every u ∈ L B (Ω; R d ), and D ∈ A (Ω) , otherwise.
Observe that the coercivity condition (H4) on f guarantees that (3.2) is equivalent at the computing the analogous limit functional with respect to the weak* convergence in Moreover, as in [18], we introduce for every u ∈ W 1 L B (Ω; R d ), Under the same assumptions of Proposition 2.6, the following result can be proven. Moreover, up to a subsequence, we may assume that where C is the set of open cubes with faces parallel to the axes, centered at x ∈ Ω ∩ Q N , with rational edge length. Let B0 ∈ R be such that C ⊂⊂ B0 ⊂⊂ B, in particular L N (∂B0) = 0. Then, by Proposition 2.6, F {εj } (u, B0) is a Γ−limit, and thus there exists a sequence ( For every u ∈ W 1 L B Ω; R d consider the functional G (u, A) := A (1 + B (|Du (x)|)) dx, and set νj k := G (uj k , ·) + G u ′ j k , · .
On the other hand, Exploiting the properties of B, we obtain Letting j → +∞ and taking into account the arbritness of 0 < α < 1 we get The difference Ω Y Z f (y, z, θj (x, y, z))dxdydz − Ω Y Z f (y, z, w0 (x, y, z))dxdydz can also be treated as done to estimate the difference of the second and third term in (3.5), thus the result follows from passing to the limit on j.  Following along the lines of [18,Theorem 4.6], now we are in position to prove the opposite inequality. Proof. Consider any subsequence of (ε), (not relabelled) such that the Γ-limit F {ε} exists. By Lemma 3.2, we know that F {ε} (u, ·) is the trace on A (Ω) of a Radon measure absolutely continous with respect to the N dimensional Lebesgue measure L N . Thus to achieve the result, it is enough to prove that for any fixed u ∈ W 1 L B Ω; R d , (Du (x0)) , for a.e x0 ∈ Ω.
Let x0 ∈ Ω, be a Lebesgue point for u, Du and assume that Fix α > 0, and using the definition of per Y ; R d and f hom is continous (see Lemma 3.1), one can take ϕ ∈ C ∞ per Y, R d such that: f hom (Du (x0))+α ≥ Y f hom (y, Du (x0) + Dϕ (y)) dy. In order to apply Proposition 2.8 with (X, M) := (Ω, L) , S := W 1 L B per Y ; R d , µ the Lebesgue measure, and L the σ−algebra of Lebesgue measurable sets in R N , we introduce the multi-valued map Exploiting the properties of Sobolev-Orlicz spaces, and the definition of f hom , we can prove that the set H (x) is non empty and closed. Indeed, let ψ1 ∈ W 1 L B per Z; R d , set ψ2 = ψ1− Z ψ1dz, we have Z ψ2dz = 0 and ∅ = H2 H2 is closed as u → Z udz is linear and continous. Next, from definition of f hom given α > 0, ∃ψ1 ∈ W 1 L B per Z; R d such that: it results that, for every n ∈ N, If ψn is such that ψ1 − ψn B → 0 as n → +∞, we get that the right hand side goes to 0, passing to limit on n and using the arbritness of 0 < α < 1, thus, due to the metrizability of the spaces, gx is continous and g −1 By Proposition 2.8 we have H (x) = {hn : n ∈ N}. Let ψ be a mesuarable selection, x ∈ Y → ψ (x, ·) ∈ W 1 L B per Z; R d , such that Z ψdz = 0 and f hom (x, Du (x0) + Dϕ (x)) + α ≥ Z f (x, y, Du (x0) + Dϕ (x) + Dyψ(x, y)) dy.
Thus, the arbritness of α gives the result.
We observe that the following Proposition, which extend to the Orlicz-Sobolev spaces, [14, lemma 4.2], can be proven. The proof develops along the lines of the above result, relying in turn on assumption (H4), approximation by means of regular functions, Lemma 2.3, and dominated convergence theorem, hence the proof is omitted. Proof of Theorem 2. It is a direct consequence of the above lemmas 3.7 and 3.6 for s = 1. For s = 2, it relies on Theorem 1.1 and minors changes with respect to the case s = 1, hence the proof is omitted.