Homogenization of nonlinear elliptic systems in nonreflexive Musielak–Orlicz spaces

Abstract

We study the homogenization process for families of strongly nonlinear elliptic systems with the homogeneous Dirichlet boundary conditions. The growth and the coercivity of the elliptic operator is assumed to be indicated by a general inhomogeneous anisotropic N -function M , which may also depend on the spatial variable, i.e., the homogenization process will change the underlying function spaces and the nonlinear elliptic operator at each step. The problem of homogenization of nonlinear elliptic systems has been solved for the L p −setting with restrictions either on constant exponent or variable exponent that is assumed to be additionally log-Hölder continuous. These results correspond to a very particular case of N -functions satisfying both ∆ 2 and ∇ 2 -conditions. We show that for general M satisfying a condition of log-Hölder type continuity, one can provide a rather general theory without any assumption on the validity of neither ∆ 2 nor ∇ 2 -conditions.

Existence and homogenization of nonlinear elliptic systems in nonreflexive spaces

“…Therefore, all results below will be always split into two cases. In case that M * satisfies ∆ 2 -condition, we just follow [2] and state all results without proofs. On the other hand, since the case when M satisfies ∆ 2 -condition is different, we provide all details for this situation.…”

“…which contradicts the unboundedness of {E(·, V n )} ∞ n=1 . For the proof of the second part of the lemma we refer to [2,Lemma 3.4. ] B Auxiliary tools Lemma B.1.…”

“…Notice that in L p(x) setting they required that 1 < p min ≤ p(x) ≤ p max < ∞, so the corresponding functions spaces were reflexive and separable as well. The first attempt to deal with the N -function not satisfying ∆ 2 -condition, was done in [2], where for the operator A fulfilling (A1)-(A4) and the function M satisfying (M1)-(M2) the limit ε → 0 was successfully established provided that M is log-Hölder continuous with respect to the first variable.…”

“…In this paper, we shall overcome this difficulty and show that even for discontinuous functions M one can obtain the fairy complete theory provided that M or M * satisfy ∆ 2 condition, but without any assumption on the continuity with respect to the spatial variable. Indeed, inspired by [22,2], we show that the limit u of a sequence of solutions to (1) satisfies a problem, in which the nonlinear operator is independent of a spatial variable, i.e., the problem possesses the form divÂ(∇u) = div F in Ω, Furthermore, assume that F ∈ L ∞ (Ω; R d×N ) (6) and for any ε > 0 let u ε be a unique solution to the problem (1). Then for an arbitrary sequence {ε j } ∞ j=1 such that ε j → 0 as j → ∞ we have the following convergence result…”

A pocket guide to nonlinear differential equations in Musielak–Orlicz spaces

“…We refer also to [107,108,110,111,196,197] for some developments arising around the theory of non-Newtonian fluids and for existence to some parabolic problems within the setting to [183,184]. Nowadays intensively investigated fields are also potential theory [115], harmonic analysis [72,121], regularity theory [116,120], and homogenization within the setting is studied in [39,40]. We want to stress embedding results of [66,152].…”