Characterization of mean value harmonic functions on norm induced metric measure spaces with weighted Lebesgue measure

TL;DR

This paper characterizes mean-value harmonic functions on weighted metric measure spaces, linking them to elliptic PDEs, and explores their properties and classifications in various geometric and analytic settings.

Contribution

It establishes necessary and sufficient conditions for mean-value harmonic functions to solve elliptic PDEs, including a complete classification in $\

Findings

For $p e 2$, only finitely many mean-value harmonic functions exist.

For $p=2$, there are infinitely many mean-value harmonic functions.

Strongly harmonic functions are analytic if the weight is analytic.

Abstract

We study the mean-value harmonic functions on open subsets of $\mathbb{R}^n$ equipped with weighted Lebesgue measures and norm induced metrics. Our main result is a necessary condition saying that all such functions solve a certain homogeneous system of elliptic PDEs. Moreover, a converse result is established in case of analytic weights. Assuming Sobolev regularity of weight $w \in W^{l,\infty}$ we show that strongly harmonic functions are as well in $W^{l,\infty}$ and that they are analytic, whenever the weight is analytic. The analysis is illustrated by finding all mean-value harmonic functions in $\mathbb{R}^2$ for the $l^p$-distance $1 \leq p \leq \infty$. The essential outcome is a certain discontinuity with respect to $p$, i.e. that for all $p \ne 2$ there are only finitely many linearly independent mean-value harmonic functions, while for $p=2$ there are infinitely many of them. We conclude with a remarkable observation that strongly harmonic functions in $\mathbb{R}^n$ possess the mean value property with respect to infinitely many weight functions obtained from a given weight.