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 CMFD, 2016, Volume 62, Pages 124–139 (Mi cmfd313)

Traces of generalized solutions of elliptic differential-difference equations with degeneration

V. A. Popov

RUDN University, 6 Miklukho-Maklaya st., 117198 Moscow, Russia

Abstract: The paper is devoted to differential-difference equations with degeneration in a bounded domain $Q\subset\mathbb R^n$. We consider differential-difference operators that cannot be expressed as a composition of a strongly elliptic differential operator and a degenerated difference operator. Instead of this, operators under consideration contain several degenerated difference operators corresponding to differentiation operators. Generalized solutions of such equations may not belong even to the Sobolev space $W^1_2(Q)$.
Earlier, under certain conditions on difference and differentiation operators, we had obtained a priori estimates and proved that the orthogonal projection of the generalized solution onto the image of the difference operator preserves certain smoothness inside some subdomains $Q_r\subset Q$ ($\bigcup_r\overline Q_r=\overline Q)$ instead of the whole domain.
In this paper, we prove necessary and sufficient conditions in algebraic form for existence of traces on some parts of boundaries of subdomains $Q_r$.

 Funding Agency Grant Number Ministry of Education and Science of the Russian Federation 1.1974.2014/K

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Citation: V. A. Popov, “Traces of generalized solutions of elliptic differential-difference equations with degeneration”, Proceedings of the Seminar on Differential and Functional Differential Equations supervised by A. L. Skubachevskii (Peoples' Friendship University of Russia), CMFD, 62, PFUR, M., 2016, 124–139

Citation in format AMSBIB
\Bibitem{Pop16} \by V.~A.~Popov \paper Traces of generalized solutions of elliptic differential-difference equations with degeneration \inbook Proceedings of the Seminar on Differential and Functional Differential Equations supervised by A.~L.~Skubachevskii (Peoples' Friendship University of Russia) \serial CMFD \yr 2016 \vol 62 \pages 124--139 \publ PFUR \publaddr M. \mathnet{http://mi.mathnet.ru/cmfd313} 

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This publication is cited in the following articles:
1. V. A. Popov, “Otsenki reshenii ellipticheskikh differentsialno-raznostnykh uravnenii s vyrozhdeniem”, Differentsialnye i funktsionalno-differentsialnye uravneniya, SMFN, 64, no. 1, Rossiiskii universitet druzhby narodov, M., 2018, 131–147
2. V. A. Popov, “Elliptic functional differential equations with degenerations”, Lobachevskii J. Math., 41:5, SI (2020), 869–894
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