Generalized solutions to the Dirichlet problem of translating mean curvature equations

2019/02/05 by Zhou, Hengyu
#. 35A01. 35J25 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Primary 35J67: Secondary 53C21. 35D 99

paper · doi:10.48550/arxiv.1902.01512

Abstract

In this paper we study the Dirichlet problem of translating mean curvature equations over domains in Riemannian manifolds with dimension n. Imitating the generalized solution theory of Miranda-Giusti, we define a new conformal area functional and a generalized solution to this Dirichlet problem. The existence of generalized solutions to this problem on bounded Lipschitz domains is established. If the domain is mean convex and bounded with C² boundary, its closure does not contain any closed minimal hypersurface except a singular set with its Hausdorff dimension at most n-7 and the boundary data is continuous, the generalized solution is the desirable classical smooth solution. The non-minimal condition could not be removed in general.

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