2022/08/15 by Zhou, Hengyu
#35J25 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Primary 53A10: Secondary 35A01
paper · doi:10.48550/arxiv.2208.07213
In this paper, we apply a blow-up method of Schoen and Yau in \citeSY81 to study a large class of prescribed mean curvature (PMC) Dirichlet problems in n(n≥ 2)-dimensional Riemannian manifolds. In this process we establish curvature estimates for almost minimizing PMC hypersurfaces, using an approach of Schauder estimates from Simon \citeSim76. 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.