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Multiple solutions to the Dirichlet problem for the equation of prescribed mean curvature. (English) Zbl 0703.53049
Analysis, et cetera, Res. Pap. in Honor of J. Moser’s 60th Birthd., 639-666 (1990).
[For the entire collection see Zbl 0688.00009.]
Any circle on \(S^ 2\) different from a great circle bounds two different surfaces of constant mean curvature. A “small” and a “large” spherical cap. Under suitable conditions on the boundary values the existence of a large solution to the Dirichlet problem and the Plateau problem for surfaces of prescribed constant mean curvature was shown only a few years ago. Here, the author succeeds in proving the same result for the Dirichlet problem if the mean curvature is not necessarily a constant itself but sufficiently near a constant. To overcome the lack of differentiability and lack of suitable a priori bounds for critical points of the associated variational problem a two-parameter family of perturbations is introduced.
Reviewer: M.Grüter

MSC:
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
58E12 Variational problems concerning minimal surfaces (problems in two independent variables)
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature