Radial Symmetry for Elliptic Boundary-Value Problems on Exterior Domains

Abstract

By the Alexandroff-Serrin method [2, 14] of moving hyperplanes we obtain radial symmetry for the domain and the solutions of on an exterior domain , subject to the overdetermined boundary conditions , on , at and in . In particular, the following conjecture from potential theory due to P. Gruber (cf. [11, 8]) is proved: Let or be a bounded smooth domain with a constant source distribution on and let be the induced single-layer potential. If is constant in , then is a ball.