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
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