Regularity of solutions to quasilinear infinitely degenerate second order equations

TL;DR

This paper proves that weak solutions to a class of infinitely degenerate quasilinear second order equations are Hölder continuous with respect to a specialized metric, advancing understanding of their regularity despite non-doubling metric balls.

Contribution

It establishes the continuity and Hölder regularity of solutions for a broad class of degenerate quasilinear equations using a novel metric framework.

Findings

Weak solutions are Hölder continuous in the specified metric.

The metric's non-doubling property is crucial for the analysis.

Results contribute to the hypoellipticity theory of degenerate elliptic operators.

Abstract

The main result of the paper is on the continuity of weak solutions of infinitely degenerate quasilinear second order equations. Namely, we show that every weak solution to a certain class of degenerate quasilinear equations is continuous. More precisely, we show that it is H\"{o}lder continuous with respect to a certain metric associated to the operator. One of the essential features of this metric is that the metric balls are non doubling with respect to Lebesgue measure. The proof of the continuity together with a recent result by Rios et al. completes the result on hypoellipticity of a class of second order quasilinear infinitely degenerate elliptic operators.