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

Abstract

We study uniqueness of $p$-harmonic Green functions in domains $\Omega$ in a complete metric space equipped with a doubling measure supporting a $p$-Poincaré 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--Véron (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 $p=2$.

References in corpus (1)

• Condenser capacities and capacitary potentials for unbounded sets, and global $p$-harmonic Green functions on metric spaces