Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
Mathematics: As an Infrastructure of Technology and Science Read chapter Read first chapter

2014 | OriginalPaper | Chapter

Stability Analysis for Variational Problems for Surfaces with Constraint

Author : Miyuki Koiso

Published in: A Mathematical Approach to Research Problems of Science and Technology

Publisher: Springer Japan

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

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.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
previous chapter Gröbner Basis and Its Applications
next chapter Discrete Models of Isoperimetric Deformation of Plane Curves
Literature
1.
J.L. Barbosa, M. do Carmo, Stability of hypersurfaces of constant mean curvature. Math. Zeit. 185, 339–353 (1984)
2.
J.L. Barbosa, M. do Carmo, J. Eschenburg, Stability of hypersurfaces of constant mean curvature in Riemannian manifolds. Math. Zeit. 197, 123–138 (1988)
3.
4.
M.G. Crandall, P.H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues, and linearized stability. Arch. Rat. Mech. Anal. 52, 161–180 (1973)MathSciNetCrossRefMATH
5.
N. Kapouleas, Complete constant mean curvature surfaces in euclidean three-space. Ann. Math. 131(2), 239–330 (1990)
6.
7.
H. Kielhöfer, in Bifurcation Theory: An Introduction with Applications to Partial Differential Equations. Applied Mathematical Sciences, 2nd edn., vol. 156. (Springer, New York, 2012)
8.
M. Koiso, Deformation and stability of surfaces with constant mean curvature. Tohoku Math. J. 54(2), 145–159 (2002)
9.
M. Koiso, B. Palmer, P. Piccione, in, Bifurcation and Symmetry Breaking of Nodoids with Fixed Boundary. Advances in Calculus of Variation (to appear)
10.
M. Koiso, B. Palmer, P. Piccione, Stability and bifurcation for surfaces with constant mean curvature (in preparation)
11.
M. Koiso, P. Piccione, T. Shoda, On bifurcation and local rigidity of triply periodic minimal surfaces in \({ R}^3\) (preprint)
12.
N. Koiso, Variational Problem (Kyoritsu, Tokyo, Japan, 1998) (In Japanese)
13.
14.
J.H. Maddocks, Restricted quadratic forms and their application to bifurcation and stability in constrained variational principles. SIAM J. Math. Anal. 16, 47–68 (1985)MathSciNetCrossRefMATH
15.
16.
U. Patnaik, Volume constrained Douglas problem and the stability of liquid bridges between two coaxial tubes. Dissertation, University of Toledo, USA, 1994
17.
18.
19.
T.I. Vogel, Stability of a liquid drop trapped between two parallel planes II. General contact angles. SIAM J. Appl. Math. 49, 1009–1028 (1989)MathSciNetCrossRefMATH
20.
T.I. Vogel, On constrained extrema. Pac. J. Math. 176, 557–561 (1996)
21.
T.I. Vogel, Sufficient conditions for multiply constrained extrema. Pac. J. Math. 180, 377–383 (1997)CrossRef
22.
T. I. Vogel, Non-linear stability of a certain capillary surfaces. Dynam. Contin. Discrete Impuls. Syst. 5, 1–15 (1999)
23.
24.
H.C. Wente, A note on the stability theorem of J. L. Barbosa and M. do Carmo for closed surfaces of constant mean curvature. Pac. J. Math. 147, 375–379 (1991)MathSciNetCrossRefMATH
Metadata
Title
Stability Analysis for Variational Problems for Surfaces with Constraint
Author
Miyuki Koiso
Copyright Year
2014
Publisher
Springer Japan
Book
A Mathematical Approach to Research Problems of Science and Technology

Print ISBN: 978-4-431-55059-4

Electronic ISBN: 978-4-431-55060-0

Copyright Year: 2014

DOI
https://doi.org/10.1007/978-4-431-55060-0_6

Premium Partners

    Image Credits