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CMFD, 2018, Volume 64, Issue 1, Pages 131–147 (Mi cmfd350)  

This article is cited in 1 scientific paper (total in 1 paper)

Estimates of solutions of elliptic differential-difference equations with degeneration

V. A. Popov

RUDN University, Moscow, Russia

Abstract: We consider a second-order differential-difference equation in a bounded domain $Q\subset\mathbb R^n$. We assume that the differential-difference operator contains some difference operators with degeneration corresponding to differentiation operators. Moreover, the differential-difference operator under consideration cannot be expressed as a composition of a difference operator and a strongly elliptic differential operator. Degenerated difference operators do not allow us to obtain the Gårding inequality.
We prove a priori estimates from which it follows that the differential-difference operator under consideration is sectorial and its Friedrichs extension exists. These estimates can be applied to study the spectrum of the Friedrichs extension as well.
It is well known that elliptic differential-difference equations may have solutions that do not belong even to the Sobolev space $W^1_2(Q)$. However, using the obtained estimates, we can prove some smoothness of solutions, though not in the whole domain $Q$, but inside some subdomains $Q_r$ generated by the shifts of the boundary, where $\bigcup_r\overline{Q_r}=\overline Q$.

Funding Agency Grant Number
Ministry of Education and Science of the Russian Federation 5-100
Russian Foundation for Basic Research 16-01-00450


DOI: https://doi.org/10.22363/2413-3639-2018-64-1-131-147

Full text: PDF file (231 kB)
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UDC: 517.9

Citation: V. A. Popov, “Estimates of solutions of elliptic differential-difference equations with degeneration”, Differential and functional differential equations, CMFD, 64, no. 1, Peoples' Friendship University of Russia, M., 2018, 131–147

Citation in format AMSBIB
\Bibitem{Pop18}
\by V.~A.~Popov
\paper Estimates of solutions of elliptic differential-difference equations with degeneration
\inbook Differential and functional differential equations
\serial CMFD
\yr 2018
\vol 64
\issue 1
\pages 131--147
\publ Peoples' Friendship University of Russia
\publaddr M.
\mathnet{http://mi.mathnet.ru/cmfd350}
\crossref{https://doi.org/10.22363/2413-3639-2018-64-1-131-147}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. V. A. Popov, “Elliptic functional differential equations with degenerations”, Lobachevskii J. Math., 41:5, SI (2020), 869–894  crossref  mathscinet  zmath  isi  scopus
  • Современная математика. Фундаментальные направления
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