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On the regularity of \(H\)-surfaces with free boundaries on a smooth support manifold. (English) Zbl 1161.53011

Author’s abstract: We study surfaces of prescribed mean curvature in \({\mathbb R}^3\) with part of their boundaries lying on a support manifold without boundary. We prove \(C^{1,\mu}\)-regularity of such a surface, whenever the support manifold is of class \(C^2\) and the surface itself is a continuous, stationary point of the associated energy functional; consequently, minimizers of that functional are included. In addition, asymptotic expansions near boundary branch points are provided. Our results improve previous work of S. Hildebrandt and W. Jäger [Math. Z. 118, 289–308 (1970; Zbl 0204.11504)] and Hardt, and generalize corresponding theorems on minimal surfaces. The main difficulty arises from the fact that stationary surfaces with prescribed mean curvature do not have to meet the support manifold perpendicularly, in contrast to minimal surfaces which are stationary points of Dirichlet’s functional.

MSC:

53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
49N60 Regularity of solutions in optimal control
49Q05 Minimal surfaces and optimization
35C20 Asymptotic expansions of solutions to PDEs

Citations:

Zbl 0204.11504
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References:

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