Abstract
We study the problem of the stability of the extremals of the potential energy functional. By the stability of an extremal surface we mean the sign-definiteness of its second variation. For estimating the second variation of the functional, we use the properties of the eigenvalues of symmetric matrices. Also, we prove an analog of Alexandrov’s Theorem on the variational property of a sphere.
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References
Hoffman D. and Osserman R., “The area of the generalized Gaussian image and the stability of minimal surfaces in \( S^{n} \) and \( ^{n} \),” Math. Ann., vol. 260, no. 4, 437–452 (1982).
Barbosa J. L. and do Carmo M., “Stability of minimal surfaces and eigenvalues of the Laplacian,” Math. Z., vol. 173, no. 1, 13–28 (1980).
Lawson H. B., “Some intrinsic characterizations of minimal surfaces,” J. Anal. Math., vol. 24, 151–161 (1971).
Pogorelov A. V., “On the stability of minimal surfaces,” Soviet Math. Dokl., vol. 24, no. 2, 274–276 (1981).
Simons J., “Minimal varieties in Riemannian manifolds,” Ann. Math., vol. 88, no. 1, 62–105 (1968).
Fomenko A. T. and Tuzhilin A. A., Elements of the Geometry of Minimal Surfaces in Three-Dimensional Space, Amer. Math. Soc., Providence (1991).
Fomenko A. T., “On the growth rate and smallest volumes of globally minimal surfaces in cobordisms,” Tr. Semin. Vekt. Tenz. Anal., vol. 21, 3–12 (1985).
Finn R., Equilibrium Capillary Surfaces, Springer, New York, Berlin, Heidelberg, and Tokyo (1986).
Saranin V. A., Equilibrium of Liquids and Its Stability, Institute of Computer Science, Moscow (2002) [Russian].
Klyachin V. A., “On some properties of stable and unstable surfaces of prescribed mean curvature,” Izv. Ross. Akad. Nauk. Ser. Mat., vol. 70, no. 4, 77–90 (2006).
Klyachin V. A. and Medvedeva N. M., “On the stability of the extremal surfaces of some area-type functionals,” Sib. Electr. Math. Reports, vol. 4, 113–132 (2007).
Poluboyarova N. M., “Equations of extremals of the functional of potential energy,” Vestn. Volgograd. Univ. Ser. 1. Mat. Fiz., vol. 36, no. 5, 60–72 (2016).
Poluboyarova N. M., “On instability of extremals of the functional of potential energy,” Ufa Math. J., vol. 10, no. 3, 77–85 (2018).
Poluboyarova N. M., “Some properties of extremals of the functional of potential energy,” Mathematical Physics, vol. 152, 103–109 (2018) (Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz.).
Alexandrov A. D., “Uniqueness theorems for surfaces in the large. V,” Amer. Math. Soc. Transl. Ser. 2, vol. 21, 412–416 (1962).
Poluboyarova N. M., “On a property of stability extremals of the functional of potential energy” in:, Complex Analysis and Approximation Theory: Abstracts of the International Conference. Ufa, May 29–31, 2019, Bashkir State Univ., Ufa (2019), 38–39.
Kobayashi Sh. and Nomizu K., Foundations of Differential Geometry. Vols. 1 and 2, Interscience, New York and London (1963).
Gelfand I. M., Lectures on Linear Algebra, Dover, New York (1989).
Funding
The author was supported by the Mathematical Center in Akademgorodok under Agreement No. 075–15–2019–1613 with the Ministry of Science and Higher Education of the Russian Federation.
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Translated from Sibirskii Matematicheskii Zhurnal, 2021, Vol. 62, No. 3, pp. 599–606. https://doi.org/10.33048/smzh.2021.62.311
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Poluboyarova, N.M. On Stable Extremals of the Potential Energy Functional. Sib Math J 62, 482–488 (2021). https://doi.org/10.1134/S0037446621030113
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DOI: https://doi.org/10.1134/S0037446621030113
Keywords
- variation of a functional
- instability of a surface
- stability of a surface
- potential energy functional
- area-type functional
- extremal surface