Green function in metric measure spaces

TL;DR

This paper investigates the existence and uniqueness of Green functions for the Cheeger Q-Laplacian in Ahlfors Q-regular metric measure spaces supporting a Q-Poincaré inequality, extending previous results to unbounded spaces.

Contribution

It establishes the existence of global Green functions in unbounded spaces and proves their uniqueness in both bounded and unbounded domains.

Findings

Uniqueness of Green functions in relatively compact domains

Existence of global Green functions in unbounded spaces

Extension of previous results to unbounded metric measure spaces

Abstract

We study existence and uniqueness of Green functions for the Cheeger $Q$-Laplacian in metric measure spaces that are Ahlfors $Q$-regular and support a $Q$-Poincar\'e inequality with $Q>1$. We prove uniqueness of Green functions both in the case of relatively compact domains, and in the global (unbounded) case. We also prove existence of global Green functions in unbounded spaces, complementing the existing results in relatively compact domains proved recently in [BBL20].