The Dirichlet problem for the minimal hypersurface equation with Lipschitz continuous boundary data on domains of a Riemannian manifold

TL;DR

This paper investigates conditions under which the Dirichlet problem for the minimal hypersurface equation can be solved on domains within Riemannian manifolds, extending previous Euclidean results to more general geometric settings.

Contribution

It extends the existence results for the minimal hypersurface equation's Dirichlet problem from Euclidean spaces to arbitrary Riemannian manifolds with Lipschitz boundary data.

Findings

Established smallness conditions on boundary data for solvability

Extended Williams' Euclidean results to Riemannian manifolds

Provided new existence theorems for minimal hypersurfaces in curved spaces

Abstract

Given a C2-domain with compact boundary in an arbitrary complete Riemannian manifold, we search for smallness conditions on the boundary data for which the Dirichlet problem for the minimal hypersurface equation is solvable. We obtain an extension to Riemannian manifolds of an existence result of G. H. Williams ( J. Reine Angew. Math. 354:123-140, 1984).