An overdetermined elliptic problem in a domain with countably rectifiable boundary
Volume 107 / 2007
Colloquium Mathematicum 107 (2007), 7-14
MSC: 35N99, 35J05, 28A99.
DOI: 10.4064/cm107-1-2
Abstract
We examine an elliptic equation in a domain ${\mit\Omega} $ whose boundary $\partial {\mit\Omega} $ is countably $(m-1)$-rectifiable. We also assume that $\partial {\mit\Omega} $ satisfies a geometrical condition. We are interested in an overdetermined boundary value problem (examined by Serrin [Arch. Ration. Mech. Anal. 43 (1971)] for classical solutions on domains with smooth boundary). We show that existence of a solution of this problem implies that ${\mit\Omega} $ is an $m$-dimensional Euclidean ball.
