Overdetermined boundary problems for maximal graphs

Abstract

We study overdetermined boundary problems for the maximal surface equation with zero Dirichlet data and constant nonzero Neumann data . For the exterior of a bounded domain with connected boundary in $\mathbb{R}^n$, $n\geq2$, we prove that such domain must be the complement of a ball and the solution is radial. In dimension two, we also show that a positive solution on a domain bounded by a properly embedded noncompact connected curve forces the domain to be a half-plane and the solution to be affine. The proofs combine the divergence structure of the equation with the induced geometry of maximal graphs.