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IMPOSING MONOTONICITY AND CURVATURE ON FLEXIBLE FUNCTIONAL FORMS
Replaced with revised version of paper 05/29/04.Research Methods/ Statistical Methods,
On the relationship of continuity and boundary regularity in PMC Dirichlet problems
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.Comment: 19 pages; typos corrected, figure added (in Figure 11), submitted to
the Pacific Journal of Mathematics; additional typos corrected. 20 pages,
accepted by the Pacific Journal of Mathematics. UPDATE: 20 pages, accepted
for publication by the Pacific Journal of Mathematic
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