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- Title
Lower semicontinuity for the Helfrich problem.
- Authors
Eichmann, Sascha
- Abstract
We minimise the Canham–Helfrich energy in the class of closed immersions with prescribed genus, surface area, and enclosed volume. Compactness is achieved in the class of oriented varifolds. The main result is a lower-semicontinuity estimate for the minimising sequence, which is in general false under varifold convergence by a counter example by Große-Brauckmann. The main argument involved is showing partial regularity of the limit. It entails comparing the Helfrich energy of the minimising sequence locally to that of a biharmonic graph. This idea is by Simon, but it cannot be directly applied, since the area and enclosed volume of the graph may differ. By an idea of Schygulla we adjust these quantities by using a two parameter diffeomorphism of R 3 .
- Subjects
SURFACE area; IMMERSIONS (Mathematics); BIHARMONIC equations
- Publication
Annals of Global Analysis & Geometry, 2020, Vol 58, Issue 2, p147
- ISSN
0232-704X
- Publication type
Article
- DOI
10.1007/s10455-020-09718-5