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 Tr. Mat. Inst. Steklova, 2014, Volume 285, Pages 64–88 (Mi tm3551)

Universal boundary value problem for equations of mathematical physics

I. V. Volovicha, V. Zh. Sakbaevb

a Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia
b Moscow Institute of Physics and Technology (State University), Dolgoprudnyi, Russia

Abstract: A new statement of a boundary value problem for partial differential equations is discussed. An arbitrary solution to a linear elliptic, hyperbolic, or parabolic second-order differential equation is considered in a given domain of Euclidean space without any constraints imposed on the boundary values of the solution or its derivatives. The following question is studied: What conditions should hold for the boundary values of a function and its normal derivative if this function is a solution to the linear differential equation under consideration? A linear integral equation is defined for the boundary values of a solution and its normal derivative; this equation is called a universal boundary value equation. A universal boundary value problem is a linear differential equation together with a universal boundary value equation. In this paper, the universal boundary value problem is studied for equations of mathematical physics such as the Laplace equation, wave equation, and heat equation. Applications of the analysis of the universal boundary value problem to problems of cosmology and quantum mechanics are pointed out.

 Funding Agency Grant Number Russian Science Foundation 14-11-00687 This work was supported by the Russian Science Foundation, project no. 14-11-00687.

DOI: https://doi.org/10.1134/S037196851402006X

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English version:
Proceedings of the Steklov Institute of Mathematics, 2014, 285, 56–80

Bibliographic databases:

UDC: 517.98

Citation: I. V. Volovich, V. Zh. Sakbaev, “Universal boundary value problem for equations of mathematical physics”, Selected topics of mathematical physics and analysis, Collected papers. In commemoration of the 90th anniversary of Academician Vasilii Sergeevich Vladimirov's birth, Tr. Mat. Inst. Steklova, 285, MAIK Nauka/Interperiodica, Moscow, 2014, 64–88; Proc. Steklov Inst. Math., 285 (2014), 56–80

Citation in format AMSBIB
\Bibitem{VolSak14} \by I.~V.~Volovich, V.~Zh.~Sakbaev \paper Universal boundary value problem for equations of mathematical physics \inbook Selected topics of mathematical physics and analysis \bookinfo Collected papers. In commemoration of the 90th anniversary of Academician Vasilii Sergeevich Vladimirov's birth \serial Tr. Mat. Inst. Steklova \yr 2014 \vol 285 \pages 64--88 \publ MAIK Nauka/Interperiodica \publaddr Moscow \mathnet{http://mi.mathnet.ru/tm3551} \crossref{https://doi.org/10.1134/S037196851402006X} \elib{https://elibrary.ru/item.asp?id=21726842} \transl \jour Proc. Steklov Inst. Math. \yr 2014 \vol 285 \pages 56--80 \crossref{https://doi.org/10.1134/S0081543814040063} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000339949700006} \elib{https://elibrary.ru/item.asp?id=24048406} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84926306257} 

• http://mi.mathnet.ru/eng/tm3551
• https://doi.org/10.1134/S037196851402006X
• http://mi.mathnet.ru/eng/tm/v285/p64

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This publication is cited in the following articles:
1. L. S. Efremova, V. Zh. Sakbaev, “Notion of blowup of the solution set of differential equations and averaging of random semigroups”, Theoret. and Math. Phys., 185:2 (2015), 1582–1598
2. V. Zh. Sakbaev, I. V. Volovich, “Self-adjoint approximations of the degenerate Schrödinger operator”, P-Adic Numbers Ultrametric Anal. Appl., 9:1 (2017), 39–52
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