Bifurcation in a variational problem on a surface with a distance constraint

Volume 7, Issue 3, pp 160--167 http://dx.doi.org/10.22436/jnsa.007.03.02

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Authors

Panayotis Vyridis - Department of Physics and Mathematics, National Polytechnical Institute (I.P.N.), Campus Zacatecas (U.P.I.I.Z.), Zacatecas, Mexico.

Abstract

We describe a variational problem on a surface of a Euclidean space under a distance constraint. We provide sufficient and necessary conditions for the existence of bifurcation points, generalizing Skrypnik's analog described in [P. Vyridis, Int. J. Nonlinear Anal. Appl. 2 (2011), 1-10]. The problem in local coordinates corresponds to an elliptic boundary value problem.

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Keywords

• Calculus of Variations
• Critical points
• Bifurcation points
• Distance function
• Curvatures of a Surface
• Boundary value problem for an elliptic PDE.

MSC

•  58E30
•  58E07
•  58E10.

References

• [1] H. Cartan, Differential Calculus - Differential Forms, Herman Paris, (1971)


• [2] B. A. Dubrovin, A. T. Fomenko, S. P. Novikov, Modern Geometry - Methods and Applications, Part II, Springer- Verlag , New York Inc. (1990)


• [3] E. Giusti, Minimal Surfaces and Functions of Bounded Variation , Monographs in Mathematics, Vol. 80 Birkhäuser, Boston-Basel-Stuttgart (1984)


• [4] V. G. Osmolovskii , Linear and nonlinear perturbations of operator div, Translations of Mathematical Monographs, Vol. 160 (1997)


• [5] I. V. Skrypnik, Nonlinear Partial Differential Equations of Higher Order, Kiev , (1973)


• [6] P. Vyridis , Bifurcation in a Variational Problem on a Surface with a Constraint, Int. J. Nonlinear Anal. Appl., 2 (2011), 1-10.


• [7] P. Vyridis, Free and Constrained Equilibrium States in a Variational Problem on a Surface , , (to appear)