On regularity of solutions to Poisson's equation

TL;DR

This paper presents new regularity results for distributional solutions to Poisson's equation, showing that their regularity is consistent across different dimensions, including solutions obtained via convolution with the fundamental solution.

Contribution

It establishes that the regularity of solutions to Poisson's equation is dimension-independent, providing new insights into their qualitative behavior.

Findings

Regularity results apply uniformly in the plane and higher dimensions.

Solutions obtained by convolution with the fundamental solution share the same regularity.

No qualitative difference in regularity across dimensions.

Abstract

In this note, we announce new regularity results for some locally integrable distributional solutions to Poisson's equation. This includes, for example, the standard solutions obtained by convolution with the fundamental solution. In particular, our results show that there is no qualitative difference in the regularity of these solutions in the plane and in higher dimensions.