Regularity of Singular Solutions to $p$-Poisson Equations

TL;DR

This paper establishes regularity results for solutions to the $p$-Poisson equation, including Lebesgue space inclusions and sharp estimates, using level set techniques for weak solutions on bounded domains.

Contribution

It introduces new level set estimates for weak solutions to the $p$-Poisson equation, leading to sharp Lebesgue space regularity results, including the critical case $q=n/p$.

Findings

Solutions belong to specific Lebesgue spaces depending on $f$'s integrability.

Regularity results are sharp and include the critical case $q=n/p$.

Level set estimates are used to derive regularity properties.

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 \[ -\Delta u=f\textrm{ in }\Omega \] \[ \quad\;\; u=0\textrm{ on }\partial\Omega \] 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 $\|u\|_r\leq C\|f\|_q$. This result is shown to be sharp, and similar regularity is established for solutions to the $p$-Poisson equation including in the edge case $q=\frac{n}{p}$.