Vol. 307, No. 1, 2020

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The Dirichlet problem for the mimimal hypersurface equation with Lipschitz continuous boundary data in a Riemannian manifold

Arì Aiolfi, Giovanni da Silva Nunes, Lisandra Sauer and Rodrigo Soares

Vol. 307 (2020), No. 1, 1–12
Abstract

In Corollary 1 of J. Reine Angew. Math. 354:123–140 (1984), G. H. Williams proves the existence of solutions to the Dirichlet problem for the minimal hypersurface equation on arbitrary bounded ${C}^{2}$ domains of the Euclidean space for Lipschitz continuous boundary data with optimal Lipschitz constant. We prove a similar result on a complete Riemannian manifold. Our theorem recovers Williams’ Corollary 1 when the ambient is the Euclidean space. Moreover, it applies to unbounded domains.

Keywords
Dirichlet problem for minimal hypersurface equation, Lipschitz continuous boundary data
Primary: 53C21
Secondary: 58J05