Symmetry results for overdetermined problems on convex domains via Brunn–Minkowski inequalities

Abstract

We consider some well-posed Dirichlet problems for elliptic equations set on the interior or the exterior of a convex domain (examples include the torsional rigidity, the first Dirichlet eigenvalue, and the electrostatic capacity), and we add an overdetermined Neumann condition which involves the Gauss curvature of the boundary. By using concavity inequalities of Brunn–Minkowski type satisfied by the corresponding variational energies, we prove that the existence of a solution implies the symmetry of the domain. This provides some new characterizations of spheres, in models going from solid mechanics to electrostatics.