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Erschienen in:
2014 | OriginalPaper | Buchkapitel
Stability Analysis for Variational Problems for Surfaces with Constraint
verfasst von : Miyuki Koiso
Erschienen in: A Mathematical Approach to Research Problems of Science and Technology
Verlag: Springer Japan
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Abstract
Surfaces with constant mean curvature (CMC surfaces) are critical points of the area functional among surfaces enclosing the same volume. Therefore, they are a simple example of solutions of variational problem with constraint. A CMC surface is said to be stable if the second variation of the area is nonnegative for all volume-preserving variations satisfying the given boundary condition. The purpose of this article is to show some fundamental methods to study the stability for CMC surfaces. Especially, we give a criterion on the stability for compact CMC surfaces with prescribed boundary. Another concept that is closely related to the stability for CMC surfaces is the so-called bifurcation. We give sufficient conditions on a one-parameter family of CMC surfaces so that there exists a bifurcation. Moreover, we give a criterion for CMC surfaces in the bifurcation branch to be stable.