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 $(n-1)\mathcal{H}_{\partial\Omega}(y)\geq n\sup\limits_{z\in\mathbb{R}}\left|{H(y,z)}\right|$ for each $y\in\partial\Omega $ provided the function $H$ also satisfies $\mathrm{Ricc}_x\geq n\sup\limits_{z\in\mathbb{R}}\left\|\nabla_x H(x,z)\right\|-\dfrac{n^2}{n-1}\inf\limits_{z\in\mathbb{R}}\left(H(x,z)\right)^2$ for each $x\in\Omega$. In the case where $M=\mathbb{H}^n$ we also establish an existence result if the condition $\sup\limits_{\Omega\times\mathbb{R}}\left|{H(x,z)}\right|\leq \frac{n-1}{n}$ 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 $-c^2\leq K\leq -1$ for some $c>1$. We generalize a classical result of Serrin when the ambient is the Euclidean space.