Regularity for quasilinear elliptic equations in metric measure spaces

TL;DR

This paper establishes second-order and Lipschitz regularity results for quasilinear elliptic equations on metric measure spaces with Ricci curvature bounds, extending classical regularity theory to a broader geometric setting.

Contribution

It introduces a novel approach using Galerkin's method to prove regularity for a wide class of elliptic equations in metric spaces, including p-harmonic functions.

Findings

Proved Lipschitz regularity for p-harmonic functions for all p in (1, ∞)

Established a Cheng-Yau type inequality in this setting

Extended regularity results beyond classical H"older theory

Abstract

In the present article we prove second-order and Lipschitz regularity for quasilinear elliptic equations in metric spaces endowed with a lower bound on the Ricci curvature. The estimates we obtain are quantitative and cover a large class of elliptic equations with polynomial growth. As a particular case we settle the Lipschitz regularity of $p$-harmonic functions for all values of $p\in(1,\infty)$, proving also a Cheng-Yau type inequality. These results are the first in this setting that simultaneously address a wide family of elliptic operators and extend beyond the classical H\"older regularity theory. Our strategy rests on the use of Galerkin's method, which we employ as an alternative to the traditional difference quotients technique.