Boundary regularity for the Poisson equation in reifenberg-flat domains
Antoine Lemenant, Yannick Sire · Scuola Normale Superiore eBooks · 2013
This paper is devoted to the investigation of the boundary regularity for the Poisson equation % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiqaaeaafa % qabeGabaaabaGaeyOeI0IaeyiLdqKaamyDaiabg2da9iaadAgacaaM % f8ocbaGaa8xAaiaa-5gacqGHPoWvaeaacaWG1bGaeyypa0JaaGimai % aaywW7caWFVbGaa8NBaiabgkGi2kabgM6axbaaaiaawUhaaaaa!4A6E! $$ \left\{ {\begin{array}{*{20}c} { - \Delta u = f\quad in\Omega } \\ {u = 0\quad on\partial \Omega } \\ \end{array} } \right. $$ where f belongs to some L p (Ω) and Ω is a Reifenberg-flat domain of ℝ N . 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, r 0)-Reifenberg-flat. The proof is based on Alt-Caffarelli-Friedman’s monotonicity formula and Morrey-Campanato theorem.