On the relationship of continuity and boundary regularity in prescribed mean curvature Dirichlet problems

Kirk E. Lancaster and Jaron Melin

Vol. 282 (2016), No. 2, 415–436

DOI: 10.2140/pjm.2016.282.415

Abstract

In 1976, Leon Simon showed that if a compact subset of the boundary of a domain is smooth and has negative mean curvature, then the nonparametric least area problem with Lipschitz continuous Dirichlet boundary data has a generalized solution which is continuous on the union of the domain and this compact subset of the boundary, even if the generalized solution does not take on the prescribed boundary data. Simon’s result has been extended to boundary value problems for prescribed mean curvature equations by other authors. In this note, we construct Dirichlet problems in domains with corners and demonstrate that the variational solutions of these Dirichlet problems are discontinuous at the corner, showing that Simon’s assumption of regularity of the boundary of the domain is essential.

Keywords

prescribed mean curvature, nonconvex corner, Dirichlet problem

Mathematical Subject Classification 2010

Primary: 76B45, 35J93

Secondary: 35J62, 53A10

Milestones

Received: 3 November 2014

Revised: 21 March 2015

Accepted: 20 July 2015

Published: 3 March 2016