An overview of regularity results for the Laplacian and $p$-Laplacian in metric spaces

TL;DR

This paper reviews regularity results for the Laplacian and p-Laplacian operators in metric measure spaces, emphasizing interior regularity estimates and conditions like Poincaré inequalities and Ricci curvature bounds.

Contribution

It provides a comprehensive overview of existing regularity results in metric spaces, highlighting key conditions and estimates for Laplacian and p-Laplacian operators.

Findings

Interior Hölder and Lipschitz regularity established

Second-regularity results discussed

Conditions like Poincaré inequality and Ricci curvature bounds are crucial

Abstract

We review some regularity results for the Laplacian and $p$-Laplacian in metric measure spaces. The focus is mainly on interior H\"older, Lipschitz and second-regularity estimates and on spaces supporting a Poincar\'e inequality or having Ricci curvature bounded below.