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arXiv · 1902.01512

Generalized solutions to the Dirichlet problem of translating mean curvature equations

Hengyu Zhou
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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 $C^2$ 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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Hengyu Zhou. 2019-02-05. Generalized solutions to the Dirichlet problem of translating mean curvature equations. https://arxiv.org/abs/1902.01512

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