Mean value formulas for solutions of some degenerate elliptic equations and applications

Abstract

We prove a mean value formula for weak solutions of $div(|y|^{a}\grad u)=0$ in ${\mathbb{R}}^{n+1}=\{(x,y):x\in {\mathbb{R}}^n,y\in \mathbb{R}\}$, $-1<a<1$ and balls centered at points of the form $(x,0)$. We obtain an explicit nonlocal kernel for the mean value formula for solutions of $(-△{)}^{s}f=0$ on a domain $D$ of ${\mathbb{R}}^n$. When $D$ is Lipschitz we prove a Besov type regularity improvement for the solutions of $(-△{)}^{s}f=0$.

arXiv comment9 pages