Sharp solvability criteria for Dirichlet problems of mean curvature type in Riemannian manifolds: non-existence results

Abstract

It is well known that the Serrin condition is a necessary condition for the solvability of the Dirichlet problem for the prescribed mean curvature equation in bounded domains of $\mathbb{R}^n$ with certain regularity. In this paper we investigate the sharpness of the Serrin condition for the vertical mean curvature equation in the product $ M^n \times \mathbb{R} $. Precisely, given a $\mathscr{C}^2$ bounded domain $\Omega$ in $M$ and a function $ H = H (x, z) $ continuous in $\overline{\Omega}\times\mathbb{R}$ and non-decreasing in the variable $z$, we prove that the strong Serrin condition $(n-1)\mathcal{H}_{\partial\Omega}(y)\geq n\sup\limits_{z\in\mathbb{R}}|H(y,z)| \ \forall \ y\in\partial\Omega $, is a necessary condition for the solvability of the Dirichlet problem in a large class of Riemannian manifolds within which are the Hadamard manifolds and manifolds whose sectional curvatures are bounded above by a positive constant. As a consequence of our results we deduce Jenkins-Serrin and Serrin type sharp solvability criteria.