RADIAL SYMMETRY FOR ELLIPTIC BOUNDARY-VALUE-PROBLEMS ON EXTERIOR DOMAINS

By the Alexandroff-Serrin method [2, 14] of moving hyperplanes we obta in radial symmetry for the domain and the solutions of Delta u + f(u, \del u\) = 0 on an exterior domain st = R-n \ <(Omega)over bar>(1), su bject to the overdetermined boundary conditions partial derivative u\p artial derivative v = const., u = const. > 0 on partial derivative Ome ga(1), u, \del u\ --> 0 at infinity, and 0 less than or equal to u < u \(partial derivative Omega 1) in Omega. In particular, the following c onjecture from potential theory due to P. GRUBER (Cf. [11, 8]) is prov ed: Let Omega(1) subset of R-2 or Omega(1) subset of R-3 be a bounded smooth domain with a constant source distribution on partial derivativ e Omega(1) and let Psi be the induced single-layer potential. If Psi i s constant in <(Omega)over bar>1, then Omega(1) is a ball.