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Ufimsk. Mat. Zh., 2021, Volume 13, Issue 1, Pages 78–85 (Mi ufa555)  

Relations between length and instability of tubular extremal surfaces

N. M. Poluboyarova

Volgograd State University, Universitetsky av. 100, 400062, Volgograd, Russia

Abstract: In the paper we study surfaces being extremals of the potential energy functional. In our case, the potential energy is the sum of two functionals, one being a functional of the area type, and the other being a functional of the volume density of forces. Extremal surfaces are stable if the second variation of the functional is sign-definite, otherwise they are unstable. In order to obtain the instability, we impose additional conditions on the surface and integrands, then we apply the properties of positive definite symmetric matrices, employ the Kronrod-Federer formula, the Cauchy-Bunyakovsky inequality, and the Weingarten homomorphism estimate. This allows us to estimate the second variation of the functional. Such technique, being a developing of an approach proposed by V.A. Klyachin, allows us to obtain conditions ensuring the instability. We establish that the length of the tubular extremal surface can be estimated in terms of the minimal and maximal $(n-1)$-dimensional measure of the cross-sections of the surface by hyperplanes. The obtained statement means that too long tubes with a non-zero mean curvature are unstable. The physical aspects of this phenomenon were considered in a work by V.A. Saranin.

Keywords: variation of a functional, extremal surface, area-type functional, volume density functional, potential energy functional, stability, instability, tubular surface, hyperplane, measure of surface section, length of tubular surface.

Funding Agency Grant Number
Ministry of Science and Higher Education of the Russian Federation 075-15-2019-1613
The work is supported by the Mathematical Center in Academgorodok, agreement with the Ministry of Science and Higher Education of Russian Federation no. 075-15-2019-1613.


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English version:
Ufa Mathematical Journal, 2021, 13:1, 77–84 (PDF, 331 kB); https://doi.org/10.13108/2021-13-1-77

Bibliographic databases:

UDC: 514.752, 514.764.274, 517.97
Received: 30.08.2020

Citation: N. M. Poluboyarova, “Relations between length and instability of tubular extremal surfaces”, Ufimsk. Mat. Zh., 13:1 (2021), 78–85; Ufa Math. J., 13:1 (2021), 77–84

Citation in format AMSBIB
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\paper Relations between length and instability of tubular extremal surfaces
\jour Ufimsk. Mat. Zh.
\yr 2021
\vol 13
\issue 1
\pages 78--85
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\jour Ufa Math. J.
\yr 2021
\vol 13
\issue 1
\pages 77--84
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