Correctors for the homogenization of almost periodic monotone operators

Abstract

In a previous paper (Braides et al., 1990) it has been proven, under very mild almost periodicity conditions, that we have weak convergence in H^1,p(Ω) of the solutions u_ε of boundary problems in an open set Ω related to the quasi-linear monotone operator −div(a(x/ε,Du_ε)), to a function u, which solves an analogous problem related to a homogenized operator −div(b(Du)).

In general we do not have strong convergence of Du_ε, to Du in (L^p(Ω))ⁿ, even in the linear periodic case.

It is possible however (Theorems 2.1 and 4.2) to express Du_ε in terms of Du, up to a rest converging strongly to 0 in (L^p(Ω))ⁿ, applying correctors built up exploiting only the geometric properties of a.

In the last section, we use the correctors result to obtain a homogenization theorem for quasi-linear equations with natural growth terms.