Vol. 297, No. 1, 2018

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ISSN: 0030-8730
Interior gradient estimates for weak solutions of quasilinear $p$-Laplacian type equations

Tuoc Phan

Vol. 297 (2018), No. 1, 195–224
Abstract

We study the interior weighted Sobolev regularity for weak solutions of the quasilinear equations of the form divA(x,u,u) = 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 < . We establish interior weighted W1,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 C1,α-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
Authors
Tuoc Phan
Department of Mathematics
University of Tennessee
Knoxville, TN
United States