Improved regularity for the $p$-Poisson equation

TL;DR

This paper establishes new optimal regularity results for solutions to the p-Poisson equation by leveraging a delicate approximation method and a stability result that connects solutions with harmonic functions, especially under a small p regime.

Contribution

It introduces a novel sequential stability technique that links p-Poisson solutions to harmonic functions, leading to improved regularity estimates.

Findings

Achieved optimal regularity results for p-Poisson solutions.

Connected solutions to harmonic functions via stability analysis.

Obtained improved low-regularity estimates under small p.

Abstract

In this paper we produce new, optimal, regularity results for the solutions to $p$-Poisson equations. We argue through a delicate approximation method, under a smallness regime for the exponent $p$, that imports information from a limiting profile driven by the Laplace operator. Our arguments contain a novelty of technical interest, namely a sequential stability result; it connects the solutions to $p$-Poisson equations with harmonic functions, yielding improved regularity for the former. Our findings relate a smallness regime with improved $\mathcal{C}^{1,1-}$-estimates in the presence of $L^\infty$-source terms.