Interior gradient estimates for weak solutions of quasilinear $p$-Laplacian type equations

Tuoc Phan

Vol. 297 (2018), No. 1, 195–224

DOI: 10.2140/pjm.2018.297.195

Abstract

We study the interior weighted Sobolev regularity for weak solutions of the quasilinear equations of the form $divA\left(x,u,\nabla u\right)=divF$. The vector field $A$ is allowed to be discontinuous in $x$, Hölder continuous in $u$ and its growth in the gradient variable is like the $p$-Laplace operator with $1<p<\infty$. We establish interior weighted $W^{1,q}$-regularity estimates for weak solutions to the equations for every $q>p$ assuming that the weak solutions are in the local John–Nirenberg $BMO$ space. This paper therefore improves available results because it replaces the boundedness or continuity assumption on weak solutions by the borderline $BMO$ one. Our regularity estimates also recover known results in which $A$ is independent of the variable $u$. Our regularity theory complements the classical $C^{1,\alpha }$-regularity theory developed by many mathematicians including DiBenedetto and Tolksdorf for this general class of quasilinear elliptic equations.

Keywords

quasilinear elliptic equations, quasilinear $p$-Laplacian type equations, Calderón–Zygmund regularity estimates, weighted Sobolev spaces

Mathematical Subject Classification 2010

Primary: 35B65, 35J62, 35J70

Secondary: 35B45

Milestones

Received: 5 May 2017

Revised: 14 January 2018

Accepted: 2 March 2018

Published: 7 October 2018