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 ${\mathcal{C}}^2$ 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 $(n-1){\mathcal{H}}_{\partial \Omega }(y)\ge n\underset{z\in \mathbb{R}}{\sup }|H(y,z)|\forall y\in \partial \Omega$, 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.

To appear in Calculus of Variations & PDE's