Boundary regularity for the Poisson equation in reifenberg-flat domains

TL;DR

This paper establishes boundary regularity results for solutions to the Poisson equation in Reifenberg-flat domains, showing local Hölder continuity under certain flatness conditions using advanced mathematical tools.

Contribution

It proves boundary Hölder regularity for the Poisson equation in Reifenberg-flat domains, extending regularity theory to less smooth boundaries.

Findings

Solutions are locally Hölder continuous in Reifenberg-flat domains.

Existence of an ε>0 ensuring regularity based on domain flatness.

Application of Alt-Caffarelli-Friedman's monotonicity formula and Morrey-Campanato theorem.

Abstract

This paper is devoted to the investigation of the boundary regularity for the Poisson equation $${{cc} -\Delta u = f & \text{in} \Omega u= 0 & \text{on} \partial \Omega$$ where $f$ belongs to some $L^p(\Omega)$ and $\Omega$ is a Reifenberg-flat domain of $\mathbb R^n.$ More precisely, we prove that given an exponent $\alpha\in (0,1)$, there exists an $\varepsilon>0$ such that the solution $u$ to the previous system is locally H\"older continuous provided that $\Omega$ is $(\varepsilon,r_0)$-Reifenberg-flat. The proof is based on Alt-Caffarelli-Friedman's monotonicity formula and Morrey-Campanato theorem.