Homogenization of p-Laplacian in perforated domain

Abstract

We study the homogenization of the following nonlinear Dirichlet variational problem:inf{∫Ωɛ{1pɛ(x)|∇u|pɛ(x)+1pɛ(x)|u|pɛ(x)−f(x)u}dx:u∈W01,pɛ(⋅)(Ωɛ)} in a perforated domain Ωɛ=Ω∖Fɛ⊂Rn, n⩾2, where ɛ is a small positive parameter that characterizes the scale of the microstructure. The non-standard exponent pɛ(x) is assumed to be an oscillating continuous function in Ω¯ such that, for any ɛ>0, 1<pɛ(x)⩽n in Ω; for any x,y∈Ω, |pɛ(x)−pɛ(y)|⩽ωɛ(|x−y|) with lim¯τ→0ωɛ(τ)ln(1/τ)=0; and converges uniformly in Ω to a function p0 which satisfies the same properties. Moreover, we assume that pɛ(x)⩾p0(x) in Ω. Denoting uɛ a minimizer in the above variational problem, without any periodicity assumption, for a large range of perforated domains we find, by means of the variational homogenization technique, the global behavior of uɛ as ɛ tends to zero. It is shown that uɛ extended by zero in Fɛ, converges weakly in W1,p0(⋅)(Ω) to the solution of the following nonlinear variational problem:min{∫Ω{1p0(x)|∇u|p0(x)+1p0(x)|u|p0(x)+c(x,u)−f(x)u}dx:u∈W01,p0(⋅)(Ω)}, where the function c(x,u) is defined in terms of the local characteristic of Ωɛ. This result is then illustrated with a periodic and a non-periodic examples.