Correctors for the homogenization of almost periodic monotone operators
Andrea Braides · Asymptotic Analysis · 1991
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.