Weighted $L^p$-estimates for elliptic equations with measurable coefficients in nonsmooth domains

TL;DR

This paper establishes weighted $L^p$ estimates for the gradients of solutions to elliptic equations with measurable coefficients in nonsmooth domains, extending regularity results to more general boundary conditions.

Contribution

It provides the first global weighted $L^p$ gradient estimates for elliptic equations with coefficients measurable in one variable and small BMO semi-norms in others, in Reifenberg flat domains.

Findings

Global weighted $L^p$ gradient estimates obtained

Solutions exhibit global Hölder continuity

Results extend regularity theory to nonsmooth domains

Abstract

We obtain a global weighted $L^p$ estimate for the gradient of the weak solutions to divergence form elliptic equations with measurable coefficients in a nonsmooth bounded domain. The coefficients are assumed to be merely measurable in one variable and to have small BMO semi-norms in the remaining variables, while the boundary of the domain is supposed to be Reifenberg flat, which goes beyond the category of domains with Lipschitz continuous boundaries. As consequence of the main result, we derive global gradient estimate for the weak solution in the framework of the Morrey spaces which implies global Hoelder continuity of the solution.