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

TL;DR

This paper establishes a mean value formula for solutions to certain degenerate elliptic equations and explores their implications for nonlocal operators like fractional Laplacians, including regularity results.

Contribution

It introduces a mean value formula for degenerate elliptic equations and derives a nonlocal kernel for fractional Laplacians, with regularity improvements on Lipschitz domains.

Findings

Derived explicit nonlocal kernel for fractional Laplacian solutions

Proved a mean value formula for degenerate elliptic equations

Established Besov regularity improvement for solutions

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 $(-\triangle)^{s}f=0$.