Homogenization of non-uniformly elliptic operators†

Abstract

An ellipticoperator A= −α _ij D _i D _j with constant coefficients is associated with any non-uniformly elliptic operator A=−D _i α _ij (x)D _j with periodic coefficients (A is called the homogenization of A), such that the solutions of Dirichlet's problems for A _ε=−D _i a _ij (xε⁻¹)D _j , converge in L ² (as ε→0) to the solution of the same problem for A. The constants α _ij can be determined by solving a differential problem relative to A. These results (which are also proved for obstacle problems) extend those obtained by several authors when A is uniformly elliptic.