Calculus of Variations and Geometric Measure Theory

A. Lemenant - Y. Sire

Boundary regularity for the Poisson equation in reifenberg-flat domains

created by lemenant on 30 Oct 2012



Inserted: 30 oct 2012
Last Updated: 30 oct 2012

Year: 2012


This paper is devoted to the investigation of the boundary regularity for the Poisson equation $$ \left\{ \begin{array}{cc} -\Delta u = f & \text{ in } \Omega \\ u= 0 & \text{ on } \partial \Omega \end{array} \right. $$ 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.