A homogenization process for the strong p(x)p(x)-Laplacian

In this work we analyze a homogenization process that takes place in a problem involving the strong p(x)p(x)-Laplacian. Precisely, we consider the following problem, closely related with Tug-of-War games {−a(x/ε)Δ∞u(x)−b(x/ε)Δu(x)=0,x∈Ω,u(x)=g(x),x∈∂Ω, with a,ba,b continuous functions such that a+b=1a+b=1 and b>0b>0 for every x∈Ωx∈Ω, being εε the homogenization parameter. Classical results in regularity theory yield the convergence of at least a subsequence of uεuε to some continuous function, as ε→0ε→0. We determine the equation verified by this limit, as well as some of its properties.