On the relationship of continuity and boundary regularity in PMC Dirichlet problems

TL;DR

This paper demonstrates that boundary regularity is crucial for the continuity of solutions in prescribed mean curvature Dirichlet problems, showing discontinuities at corners where boundary smoothness fails.

Contribution

It constructs examples in domains with corners where variational solutions are discontinuous, highlighting the importance of boundary regularity assumptions.

Findings

Solutions are discontinuous at corners in non-smooth domains.

Boundary regularity is essential for solution continuity in PMC Dirichlet problems.

Discontinuities occur where boundary smoothness assumptions are violated.

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 non-parametric 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.