Application of Newton's method to a homogenization problem

Abstract

The homogenization of a family (P_ε) of uniformly elliptic semilinear partial differential equations of second order is studied. The main result is that any non-singular solutionu of the homogenized problem (P) is the limit of non-singular solutions of (P_ε). The method consists of specifying a functionw ε starting from which the Newton iterates converge to a solutionu u_ε of P_ε. These solutions u_ε converge to the given solution u of (P).