Harnack's inequality and H\"older continuity for weak solutions of degenerate quasilinear equations with rough coefficients

TL;DR

This paper establishes Harnack's inequality and H"older continuity for weak solutions of degenerate quasilinear equations with rough coefficients, extending classical results to more general, less smooth settings.

Contribution

It introduces new regularity results for degenerate quasilinear equations with minimal smoothness assumptions, including a version of Harnack's inequality and local H"older continuity.

Findings

Proves Harnack's inequality for weak solutions.

Establishes local H"older continuity of solutions.

Extends classical regularity results to equations with rough coefficients.

Abstract

We continue to study regularity results for weak solutions of the large class of second order degenerate quasilinear equations of the form \begin{eqnarray} \text{div}\big(A(x,u,\nabla u)\big) = B(x,u,\nabla u)\text{ for }x\in\Omega\nonumber \end{eqnarray} as considered in our previous paper giving local boundedness of weak solutions. Here we derive a version of Harnack's inequality as well as local H\"older continuity for weak solutions. The possible degeneracy of an equation in the class is expressed in terms of a nonnegative definite quadratic form associated with its principal part. No smoothness is required of either the quadratic form or the coefficients of the equation. Our results extend ones obtained by J. Serrin and N. Trudinger for quasilinear equations, as well as ones for subelliptic linear equations obtained by Sawyer and Wheeden in their 2006 AMS memoir article.