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 C2 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
Mathematical Subject Classification 2010
Primary: 53C21
Secondary: 58J05
Milestones
Received: 11 September 2018
Revised: 27 January 2020
Accepted: 27 March 2020
Published: 8 August 2020
Authors
Arì Aiolfi
Departamento de Matemática
Universidade Federal de Santa Maria
Santa Maria
Brazil
Giovanni da Silva Nunes
Instituto de Física e Matemática
Universidade Federal de Pelotas
Pelotas
Brazil
Lisandra Sauer
Instituto de Física e Matemática
Universidade Federal de Pelotas
Pelotas
Brazil
Rodrigo Soares
Instituto de Matemática e Estatística
Universidade Federal do Rio Grande
Rio Grande
Brazil