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 Ufimsk. Mat. Zh., 2018, Volume 10, Issue 3, Pages 79–88 (Mi ufa440)

On instability of extremals of potential energy functional

N. M. Poluboyarova

Abstract: The paper is devoted to studying the stability and instability of extremals of a potential energy functional. A particular case of this functional is the area type functionals. The potential energy functional is the sum of functionals of area type and of volume density of forces. The potential energy functional is constructed in such way in order to take into consideration the loads on the surface from outside and inside. The stability is defined as the sign-definiteness of the second variation. In this paper we prove the formulae for the first and second variations of the functional. We also prove that the extremal surface can be locally minimal and locally maximal depending on the sign of matrix $G$. Using the $G$-capacity and the second variation of the functional, we obtain the conditions for the instability of the extremals of the potential energy functional. This technique was developed in works by V.M. Miklyukov and V.A. Klyachin. For $G$-parabolic extremal surfaces we prove the degeneracy into the plane. This result is an analogue of the theorems by M. do Carmo and C.K. Peng. By an example of $n$-dmensional surfaces of revolution we demonstrate the formulae for the first and second variations of the functional. We also prove the criteria of stability and instability for $n$-dimensional surfaces of revolution. Similar extremal surfaces arise in applications, in physical problems (e.g. soap films, capillary surfaces, magnetic liquids in a gravitational field with a potential), and the properties of extreme surfaces are used in applied problems (e.g. modeling of awning coverings).

Keywords: the variation of functional, extreme surface, area type functional, volumetric power density functional, functional of potential energy, $G$-capacity, $G$-parabolicity, the stability.

 Funding Agency Grant Number Russian Foundation for Basic Research 15-41-02479_ð_ïîâîëæüå_à The study is supported by RFBR (grant no. 15-41-02479 r).

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English version:
Ufa Mathematical Journal, 2018, 10:3, 77–85 (PDF, 350 kB); https://doi.org/10.13108/2018-10-3-77

Bibliographic databases:

UDC: 514.752, 514.764.274, 517.97
MSC: 53A10, 30C70, 31A15

Citation: N. M. Poluboyarova, “On instability of extremals of potential energy functional”, Ufimsk. Mat. Zh., 10:3 (2018), 79–88; Ufa Math. J., 10:3 (2018), 77–85

Citation in format AMSBIB
\Bibitem{Pol18} \by N.~M.~Poluboyarova \paper On instability of extremals of potential energy functional \jour Ufimsk. Mat. Zh. \yr 2018 \vol 10 \issue 3 \pages 79--88 \mathnet{http://mi.mathnet.ru/ufa440} \transl \jour Ufa Math. J. \yr 2018 \vol 10 \issue 3 \pages 77--85 \crossref{https://doi.org/10.13108/2018-10-3-77} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000457365400006} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85057013659} 

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This publication is cited in the following articles:
1. N. M. Poluboyarova, “Relations between length and instability of tubular extremal surfaces”, Ufa Math. J., 13:1 (2021), 77–84
2. N. M. Poluboyarova, “Ob ustoichivykh ekstremalyakh funktsionala potentsialnoi energii”, Sib. matem. zhurn., 62:3 (2021), 595–602
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