A blow-up method to prescribed mean curvature graphs with fixed boundaries

Abstract

In this paper, we apply a blow-up method of Schoen and Yau in \cite{SY81} to study a large class of prescribed mean curvature (PMC) Dirichlet problems in $n(n\geq 2)$-dimensional Riemannian manifolds. In this process we establish curvature estimates for almost minimizing PMC hypersurfaces, using an approach of Schauder estimates from Simon \cite{Sim76}. We define an Nc-f domain, where $f$ is a given function generating from the PMC equation. Combining this condition with a sufficiently mean convex assumption the blow-up method yields corresponding solutions to these PMC Dirichlet problems. Such Nc-f assumption is almost optimal by an example. An application of our result into the PMC Plateau problem is also presented.