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

DOI: 10.2140/pjm.2020.307.1

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

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