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 $(-\triangle)^{s}f=0$ on a domain $D$ of $\mathbb{R}^{n}$. When $D$ is Lipschitz we prove a Besov type regularity improvement for the solutions of $(-\triangle)^{s}f=0$.