Existence Serrin type results for the Dirichlet problem for the prescribed mean curvature equation in Riemannian manifolds

Abstract

Given a complete $n$-dimensional Riemannian manifold $M$, we study the existence of vertical graphs in $M\times \mathbb{R}$ with prescribed mean curvature $H=H(x,z)$. Precisely, we prove that the Dirichlet problem for the vertical mean curvature equation in a smooth bounded domain $\Omega \subset M$ has solution for arbitrary smooth boundary data if $(n-1){\mathcal{H}}_{\partial \Omega }(y)\ge n\underset{z\in \mathbb{R}}{\sup }\left|H(y,z)\right|$ for each $y\in \partial \Omega$ provided the function $H$ also satisfies ${\mathrm{Ricc}}_x\ge n\underset{z\in \mathbb{R}}{\sup }\Vert {\nabla }_{x}H(x,z)\Vert -\frac{n^2}{n-1}\underset{z\in \mathbb{R}}{\inf }{\left(H(x,z)\right)}^2$ for each $x\in \Omega$. In the case where $M={\mathbb{H}}^n$ we also establish an existence result if the condition $\underset{\Omega \times \mathbb{R}}{\sup }\left|H(x,z)\right|\le \frac{n-1}{n}$ holds in the place of the condition involving the Ricci curvature. Finally, we have a related result when $M$ is a Hadamard manifold whose sectional curvature $K$ satisfies $-c^2\le K\le -1$ for some $c>1$. We generalize a classical result of Serrin when the ambient is the Euclidean space.

improved version