Boundary regularity for quasilinear elliptic equations with general Dirichlet boundary data

TL;DR

This paper establishes global gradient regularity estimates for solutions to quasilinear elliptic equations with rough boundary data in irregular domains, extending regularity theory to more general settings.

Contribution

It provides new global gradient estimates in weighted Morrey spaces for solutions in Reifenberg flat domains with minimal boundary regularity assumptions.

Findings

Global gradient estimates in weighted Morrey spaces

Applicable to Reifenberg flat domains with minimal boundary regularity

Extends regularity results to nonhomogeneous boundary data

Abstract

We study global regularity for solutions of quasilinear elliptic equations of the form $\div \A(x,u,\nabla u) = \div \F $ in rough domains $\Omega$ in $\R^n$ with nonhomogeneous Dirichlet boundary condition. The vector field $\A$ is assumed to be continuous in $u$, and its growth in $\nabla u$ is like that of the $p$-Laplace operator. We establish global gradient estimates in weighted Morrey spaces for weak solutions $u$ to the equation under the Reifenberg flat condition for $\Omega$, a small BMO condition in $x$ for $\A$, and an optimal condition for the Dirichlet boundary data.