Harmonic functions via restricted mean-value theorems

Let $f$ be a function on a bounded domain $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ and $\delta$ be a positive function on $\mathrm{\Omega }$ such that $B(x,\delta(x))\subseteq \Omega$\mathrm{.}Let$\sigma(f)(x)$betheaverageof$f$overtheball$B(x,\delta(x))$\mathrm{.}Therestrictedmean-valuetheoremsdiscusstheconditionson$f,\delta,$and$\Omega$underwhich$\sigma(f)=f$impliesthat$f$ 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 $\mathrm{\Omega }$ 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$\sigma^n(f)$convergestoaharmonicfunctionuniformlyon$\bar \Omega$.