# A short proof of local regularity of distributional solutions of Poisson's equation

## TL;DR

This paper presents a concise proof demonstrating local regularity of distributional solutions to Poisson's equation with L^p data, utilizing Weyl's lemma and Riesz-Fréchet theorem.

## Contribution

It introduces a very short and elegant proof of local regularity for solutions of Poisson's equation with L^p data, simplifying previous approaches.

## Key findings

- Established local regularity for distributional solutions with minimal assumptions.
- Provided a concise proof leveraging classical functional analysis tools.
- Enhanced understanding of regularity properties in elliptic PDEs.

## Abstract

We prove a local regularity result for distributional solutions of the Poisson's equation with $L^p$ data. We use a very short argument based on Weyl's lemma and Riesz-Fr\'echet representation theorem.