Harmonic functions via restricted mean-value theorems
| dc.creator | Javaheri, Mohammad | |
| dc.date | 2007-09-20 | |
| dc.date.accessioned | 2026-07-07T08:31:20Z | |
| dc.date.available | 2026-07-07T08:31:20Z | |
| dc.description | Let $f$ be a function on a bounded domain $Ω\subseteq \mathbb{R}^n$ and $δ$ be a positive function on $Ω$ such that $B(x,δ(x))\subseteq Ω$. Let $σ(f)(x)$ be the average of $f$ over the ball $B(x,δ(x))$. The restricted mean-value theorems discuss the conditions on $f,δ,$ and $Ω$ under which $σ(f)=f$ implies that $f$ is harmonic. In this paper, we study the stability of harmonic functions with respect to the map $σ$. One expects that, in general, the sequence $σ^n(f)$ converges to a harmonic function. Among our results, we show that if $Ω$ is strongly convex (respectively $C^{2,α}$-smooth for some $α\in [0,1]$), the function $δ(x)$ is continuous, and $f\in C^0(\bar Ω)$ (respectively, $f\in C^{2,α}(\bar Ω)$), then $σ^n(f)$ converges to a harmonic function uniformly on $\bar Ω$. | |
| dc.description | 9 pages | |
| dc.identifier | https://arxiv.org/abs/0709.3311 | |
| dc.identifier | http://arxiv.org/abs/0709.3311 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/138477 | |
| dc.subject | Analysis of PDEs | |
| dc.subject | 65N99, 35J25, 34K10 | |
| dc.title | Harmonic functions via restricted mean-value theorems | |
| dc.type | text |