Sharp regularity for general Poisson equations with borderline sources

TL;DR

This paper establishes optimal regularity estimates for solutions to degenerate elliptic equations with borderline integrability sources, showing bounded mean oscillation and exponential integrability in rough media.

Contribution

It provides sharp regularity results for degenerate elliptic equations with sources in borderline weak Lebesgue spaces, extending previous understanding of solution behavior.

Findings

Solutions are Hölder continuous with sharp estimates.

Solutions with borderline sources have bounded mean oscillation.

Results are optimal and applicable to rough media.

Abstract

This article concerns optimal estimates for non-homogeneous degenerate elliptic equation with source functions in borderline spaces of integrability. We deliver sharp H\"older continuity estimates for solutions to $p$-degenerate elliptic equations in rough media with sources in the weak Lebesgue space $L_\text{weak}^{\frac{n}{p} + \epsilon}$. For the borderline case, $f \in L_\text{weak}^{\frac{n}{p}}$, solutions may not be bounded; nevertheless we show that solutions have bounded mean oscillation, in particular John-Nirenberg's exponential integrability estimates can be employed. All the results presented in this paper are optimal. Our approach is based on powerful Caffarelli-type compactness methods and it can be employed in a number order situations.