Homogenization of a class of quasilinear elliptic equations with non-standard growth in high-contrast media

Abstract

We study the asymptotic behaviour of solutions to a quasilinear equation with high-contrast coefficients. The energy formulation of the problem leads to work with variable exponent Lebesgue spaces L^{p_ε (·)} in a domain Ω with a complex microstructure depending on a small parameter ε. Assuming only that the functions p_ε converge uniformly to a limit function p₀ and that p₀ satisfy certain logarithmic uniform continuity conditions, we rigorously derive the corresponding homogenized problem which is completely described in terms of local energy characteristics of the original domain. In the framework of our method we do not have to specify the geometrical structure Ω. We illustrate our result with periodical examples, extending, in particular, the classical extension results to variable exponent Sobolev spaces.