Lipschitz continuity of solutions of Poisson equations in metric measure spaces

TL;DR

This paper investigates the Lipschitz regularity of solutions to Poisson equations in metric measure spaces with specific geometric and measure-theoretic properties, using heat equation techniques.

Contribution

It establishes local Lipschitz continuity of solutions when the source term is in L^p with p>Q, under certain geometric and functional inequalities.

Findings

Solutions are locally Lipschitz continuous for p>Q.

Uses heat equation methods to analyze regularity.

Extends regularity results to metric measure spaces with Ahlfors regularity.

Abstract

Let $(X,d)$ be a pathwise connected metric space equipped with an Ahlfors $Q$-regular measure $\mu$, $Q\in[1,\infty)$. Suppose that $(X,d,\mu)$ supports a 2-Poincar\'e inequality and a Sobolev-Poincar\'e type inequality for the corresponding "Gaussian measure". The author uses the heat equation to study the Lipschitz regularity of solutions of the Poisson equation $\Delta u=f$, where $f\in L^p_\loc$. When $p>Q$, the local Lipschitz continuity of $u$ is established.