Boundary regularity for the Poisson equation in reifenberg-flat domains

2012/10/31 by Lemenant, Antoine, Sire, Yannick · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1210.8305

Abstract

This paper is devoted to the investigation of the boundary regularity for the Poisson equation cc -Δu = f amp; in Ω u= 0 amp; on ∂ Ω where f belongs to some L^p(Ω) and Ω is a Reifenberg-flat domain of \mathbb Rⁿ. More precisely, we prove that given an exponent α∈ (0,1), there exists an ε>0 such that the solution u to the previous system is locally Hölder continuous provided that Ω is (ε,r₀)-Reifenberg-flat. The proof is based on Alt-Caffarelli-Friedman's monotonicity formula and Morrey-Campanato theorem.

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• Quantitative boundary Hölder estimates for the inhomogeneous Poisson problem through a probabilistic approach

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