Some Properties of Extremals of the Functional of Potential Energy
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SOME PROPERTIES OF EXTREMALS OF THE FUNCTIONAL OF POTENTIAL ENERGY N. M. Poluboyarova
UDC 514.752, 514.764.274, 517.97
Abstract. In this paper, we discuss stability and instability criteria for extremal surfaces of a special functional, which is a linear combination of an area-type functional and the functional of volumetric force density. Extremals of such functionals can serve as models of physically equilibrium tents or liquids in potential gravitational fields, so that the problem of their stability or instability is very topical. Our results are based on various geometric properties of surfaces; they are obtained by methods developed by V. M. Miklyukov and V. A. Klyachin. Keywords and phrases: variation of functional, extremal surface, extremal of surface, area-type functional, functional of volumetric force density, functional of potential energy, stability, instability, Jacobi equation, base frequency. AMS Subject Classification: 53A10, 30C70, 31A15
1. Introduction. It is known that the establishment of stability (instability) conditions is an important problem in the theory of minimal surfaces. We treat the stability as the sign-definiteness of the second variation of a special functional under any infinitesimal deformations of a surface M. Since the pioneer studies of minimal surfaces go back to Lagrange (1768) and continue up to this day, we explain the line of research development. Extremal surfaces can serve as models of states of equilibrium liquids in a potential gravitational field, awnings, magnetic liquids, capillary surfaces, etc. Therefore, the study of their stability and instability only changes in the form of functionals in order to accommodate more physical characteristics of the system. For example, a functional (e.g., energy) can be a combination of the surface tension energy, the gravitational energy, and the bending strain energy (see [14]). In this paper, we analyze the stability of extremals of the functional W (M) = Φ(ξ) dM + Ψ(x) dx, M
Ω
which is a linear combination of an area-type functional and the functional of volumetric force density; we call this functional the functional of potential energy. Here M is a C 2 -smooth surface, ξ is the field of unit normals to M, Ω ⊂ Rn+1 is a domain such that M ⊂ ∂Ω, and Φ, Ψ : Rn+1 → R are C 2 -smooth functions. The study of this functional was based on results obtained earlier in [9] for the area-type functional F (M) = Φ(ξ)dM, M
which plays an important role in applications. In the study of capillary surfaces, functionals with a nonlinear function depending on the unit normal to the surface appear; these functionals differ from the volume functionals. In particular, R. Finn in [3] examined the stability of capillary surfaces, and V. A. Saranin in [16] examined the stability of so-called magnetic fluids; these problems led to the Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 152, Mathematical Physics, 2018. c 2021 Springer Science+Business Media, LLC 1072–3374/21/2522
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