Regularity estimates in weighted Morrey spaces for quasilinear elliptic equations

TL;DR

This paper establishes interior gradient regularity estimates in weighted Morrey spaces for solutions to quasilinear elliptic equations with growth similar to the p-Laplace operator, under small BMO conditions.

Contribution

It introduces new weighted Morrey space regularity results for solutions of quasilinear elliptic equations with minimal assumptions on coefficients.

Findings

Gradient estimates in weighted Morrey spaces for weak solutions.

Regularity transfer from data to solution's gradient in weighted Morrey spaces.

Two-weight estimates allowing different regularity spaces for data and solutions.

Abstract

We study regularity for solutions of quasilinear elliptic equations of the form $\div \A(x,u,\nabla u) = \div \F $ in bounded domains in $\R^n$. 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 interior gradient estimates in weighted Morrey spaces for weak solutions $u$ to the equation under a small BMO condition in $x$ for $\A$. As a consequence, we obtain that $\nabla u$ is in the classical Morrey space $\calM^{q,\lambda}$ or weighted space $L^q_w$ whenever $|\F|^{\frac{1}{p-1}}$ is respectively in $\calM^{q,\lambda}$ or $L^q_w$, where $q$ is any number greater than $p$ and $w$ is any weight in the Muckenhoupt class $A_{\frac{q}{p}}$. In addition, our two-weight estimate allows the possibility to acquire the regularity for $\nabla u$ in a weighted Morrey space that is different from the functional space that the data $|\F|^{\frac{1}{p-1}}$ belongs to.