Symmetry for exterior elliptic problems and two conjectures in potential theory

In this paper we extend a classical result of Serrin to a class of elliptic problems Deltau + f (u, \ delu \) = 0 in exterior domains R-N \ G (or Omeg a \G with Omega and G bounded). In case G is an union of a finite number of disjoint C-2-domains G(i) and u = a(i) > 0, partial derivativeu/partial de rivativen = alpha (i) less than or equal to 0 on partial derivativeG(i), u --> 0 at infinity, we show that if a non-negative solution of such a proble m exists, then G has only one component and it is a ball. As a consequence we establish two results in electrostatics and capillarity theory. We furth er obtain symmetry results for quasilinear elliptic equations in the exteri or of a ball. (C) 2001 Editions scientifiques et medicales Elsevier SAS.