OnW1,p estimates for elliptic equations in divergence form

Abstract

First, we will apply the method to study W 1,p regularity for a nonlinear elliptic operator in divergence form. We would like to point out that in the particular case of a linear elliptic equation, this method gives an alternative proof to the classical one, which uses the general theory of singular integrals. The utility of the method that we describe below is that hypotheses (A) and (B) are obtained directly by studying the deviation of the “coefficients” of A from the “coefficients” of A0, and this is usually not a difficult task. A more interesting application of this kind of approximation method is to elliptic homogenization problems for which we obtain results that give W 1,p estimates with a weak hypothesis of regularity in the coefficients. For instance, we are able to study the following cases: