Uniqueness and nonuniqueness of $p$-harmonic Green functions on weighted $\mathbf{R}^n$ and metric spaces

TL;DR

This paper investigates the conditions under which $p$-harmonic Green functions are unique or non-unique in various metric space settings, extending known results and providing new examples of non-uniqueness.

Contribution

It establishes a sufficient condition for the uniqueness of $p$-harmonic Green functions in metric spaces and presents the first example of non-uniqueness even for $p=2$.

Findings

Uniqueness holds when the singularity has positive $p$-capacity.

A nondegenerate interval of $p$ values ensures uniqueness.

Examples show non-uniqueness can occur even at $p=2$.

Abstract

We study uniqueness of $p$-harmonic Green functions in domains $\Omega$ in a complete metric space equipped with a doubling measure supporting a $p$-Poincar\'e inequality, with $1<p<\infty$. For bounded domains in unweighted $\mathbf{R}^n$, the uniqueness was shown for the $p$-Laplace operator $\Delta_p$ and all $p$ by Kichenassamy--V\'eron (Math. Ann. 275 (1986), 599-615), while for $p=2$ it is an easy consequence of the linearity of the Laplace operator $\Delta$. Beyond that, uniqueness is only known in some particular cases, such as in Ahlfors $p$-regular spaces, as shown by Bonk--Capogna--Zhou (arXiv:2211.11974). When the singularity $x_0$ has positive $p$-capacity, the Green function is a particular multiple of the capacitary potential for $\text{cap}_p(\{x_0\},\Omega)$ and is therefore unique. Here we give a sufficient condition for uniqueness in metric spaces, and provide an example showing that the range of $p$ for which it holds (while $x_0$ has zero $p$-capacity) can be a nondegenerate interval. In the opposite direction, we give the first example showing that uniqueness can fail in metric spaces, even for $p=2$.