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Stability Analysis for Variational Problems for Surfaces with Constraint

Chapter
Part of the Mathematics for Industry book series (MFI, volume 5)

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.

Keywords

Bifurcation Constant mean curvature Pitchfork bifurcation Stability Symmetry breaking Variational problem 

Notes

Acknowledgments

The author is partially supported by Grant-in-Aid for Scientific Research (B) No. 25287012 of the Japan Society for the Promotion of Science, and the Kyushu University Interdisciplinary Programs in Education and Projects in Research Development.

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Copyright information

© Springer Japan 2014

Authors and Affiliations

  1. 1.Institute of Mathematics for IndustryKyushu UniversityNishi-kuJapan

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