The Helfrich boundary value problem

We construct a branched Helfrich immersion satisfying Dirichlet boundary conditions. The number of branch points is finite. We proceed by a variational argument and hence examine the Helfrich energy for oriented varifolds. The main contribution of this paper is a lower semicontinuity result with res...

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Bibliographic Details
Published in:Calculus of variations and partial differential equations 2019-02, Vol.58 (1), p.1-26, Article 34
Main Author: Eichmann, Sascha
Format: Article
Language:English
Subjects:
Analysis
Boundary conditions
Boundary value problems
Calculus of Variations and Optimal Control
Optimization
Control
Dirichlet problem
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Submerging
Systems Theory
Theoretical
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Summary:We construct a branched Helfrich immersion satisfying Dirichlet boundary conditions. The number of branch points is finite. We proceed by a variational argument and hence examine the Helfrich energy for oriented varifolds. The main contribution of this paper is a lower semicontinuity result with respect to oriented varifold convergence for the Helfrich energy and a minimising sequence. For arbitrary sequences this is false by a counterexample of Große-Brauckmann.
ISSN:0944-2669
1432-0835
DOI:10.1007/s00526-018-1468-x