  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.
