Harmonic functions via restricted mean-value theorems

Abstract

Let $f$ be a function on a bounded domain $\Omega \subseteq {\mathbb{R}}^n$ and $\delta$ be a positive function on $\Omega$ such that $B(x,\delta (x))\subseteq \Omega$. Let $\sigma (f)(x)$ be the average of $f$ over the ball $B(x,\delta (x))$. The restricted mean-value theorems discuss the conditions on $f,\delta ,$ and $\Omega$ under which $\sigma (f)=f$ implies that $f$ is harmonic. In this paper, we study the stability of harmonic functions with respect to the map $\sigma$. One expects that, in general, the sequence ${\sigma }^n(f)$ converges to a harmonic function. Among our results, we show that if $\Omega$ is strongly convex (respectively $C^{2,\alpha }$-smooth for some $\alpha \in [0,1]$), the function $\delta (x)$ is continuous, and $f\in C^0(\bar{\Omega })$ (respectively, $f\in C^{2,\alpha }(\bar{\Omega })$), then ${\sigma }^n(f)$ converges to a harmonic function uniformly on $\bar{\Omega }$.

9 pages