Regularity of Singular Solutions to $p$-Poisson Equations

Abstract

This work showcases level set estimates for weak solutions to the $p$-Poisson equation on a bounded domain, which we use to establish Lebesgue space inclusions for weak solutions. In particular we show that if $\Omega \subset {\mathbb{R}}^n$ is a bounded domain and $u$ is a weak solution to the Dirichlet problem for Poisson's equation \[ -Δu=f\textrm{ in }Ω\] \[ \quad\;\; u=0\textrm{ on }\partialΩ\] for $f\in L^q(\Omega )$ with $q<\frac{n}{2}$, then $u\in L^r(\Omega )$ for every $r<\frac{qn}{n-2q}$ and indeed $\Vert u{\Vert }_r\le C\Vert f{\Vert }_q$. This result is shown to be sharp, and similar regularity is established for solutions to the $p$-Poisson equation including in the edge case $q=\frac{n}{p}$.

7 pages