Elliptic Equations With Degenerate weights

TL;DR

This paper develops new local Calderon-Zygmund estimates for elliptic equations with matrix-valued weights, including degenerate and discontinuous cases, by introducing a novel log-BMO condition on the weights.

Contribution

It introduces a new log-BMO condition on matrix weights, enabling sharp estimates for degenerate elliptic equations with discontinuous weights.

Findings

Established local Calderon-Zygmund estimates under the log-BMO condition.

Included examples demonstrating the sharpness of the estimates.

Extended the theory to encompass degenerate, discontinuous weights.

Abstract

We obtain new local Calderon-Zygmund estimates for elliptic equations with matrix-valued weights for linear as well as non-linear equations. We introduce a novel log-BMO condition on the weight M. In particular, we assume smallness of the logarithm of the matrix-valued weight in BMO. This allows to include degenerate, discontinuous weights. We provide examples that show the sharpness of the estimates in terms of the log-BMO-norm.