Homogenization Of A Class Of Nonlinear Elliptic Problems

We study the existence, uniqueness and asymptotic behavior, as ∈â†'> 0, of solution of nonlinear elliptic problem where α∈(x, â–½u∈)= a(x/∈,â–½u∈), a:Rn×Rnâ†'Rn is Aperiodic to the first variable, and is a monotone, Lipschitz continuous operator with respect to the second one, moreover a(x,0)= 0 for a.e. x ∈Rn.f:Ω×Râ†'R is a Caratheodory function,{∈} is a sequence tending to 0. We give some conditions on f to ensure the existence and uniqueness of solution, and will see that they also ensure that{u∈} is uniformly bounded in H01(Ω) Naturally, We consider the behavior of weak limit of subsequence of{u∈} and prove that the weak limit is the weak solution of homogenization problem. In this process, we mainly use the TarTar’s method and compensated compactness lemma.