Correctors for Some Nonlinear Monotone Operators

In this paper we study homogenization of quasi-linear partial differential equations of the form −div (a (x, x/εh, Duh)) = fh on Ω with Dirichlet boundary conditions. Here the sequence (εh) tends to 0 as h → ∞ and the map a (x, y, ξ) is periodic in y, monotone in ξ and satisfies suitable continuity conditions. We prove that uh → u weakly in W 1,p 0 (Ω) as h → ∞, where u is the solution of a homogenized problem of the form −div (b (x,Du)) = f on Ω. We also derive an explicit expression for the homogenized operator b and prove some corrector results, i.e. we find (Ph) such that Duh − Ph (Du) → 0 in L (Ω,R).