Correctors for the homogenization of almost periodic monotone operators

Article type: Research Article

Authors: Braides, Andrea

Affiliations: Dipartimento Automazione Industriale, Università degli Studi di Brescia, Via Valotti 9, I-25060 Brescia, Italy

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 H1,p(Ω) of the solutions uε of boundary problems in an open set Ω related to the quasi-linear monotone operator −div(a(x/ε,Duε)), 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 Duε, to Du in (Lp(Ω))n, even in the linear periodic case. It is possible however (Theorems 2.1 and 4.2) to express Duε in terms of Du, up to a rest converging strongly to 0 in (Lp(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.

DOI: 10.3233/ASY-1991-5103

Journal: Asymptotic Analysis, vol. 5, no. 1, pp. 47-74, 1991