RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PERSONAL OFFICE
 General information Latest issue Forthcoming papers Archive Impact factor Subscription Guidelines for authors License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Mat. Sb.: Year: Volume: Issue: Page: Find

 Personal entry: Login: Password: Save password Enter Forgotten password? Register

 Mat. Sb., 1995, Volume 186, Number 2, Pages 37–58 (Mi msb12)

This article is cited in 16 scientific papers (total in 16 papers)

On the continuity of the solutions of a class of non-local problems for an elliptic equation

A. K. Gushchina, V. P. Mikhailov

a Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: This paper is devoted to a study of the connection between the notion of an $(n-1)$-dimensionally continuous (weak) solution for a non-local problem, which was earlier introduced by the authors, with the notion of a classical solution. Under natural suppositions on the operator entering the non-local condition, the continuity of the weak solution in the closure of the domain under consideration is proved for all arbitrary continuous boundary function. The notion of an $(n-1)$-dimensionally continuous solution is convenient when studying the Fredholm property of the problem. In the previous paper of the authors tl.e Fredholm property in such a setting was proved for a wide class of non-local problems. When studying the uniqueness it is easier to deal with a classical solution. The main result of this paper enables one, in particular, to use simultaneously the advantages of both approaches: to apply the classical maximum principle in the proof of the uniqueness (and hence, by the Fredholm property, the existence) of a weak solution.

Full text: PDF file (2361 kB)
References: PDF file   HTML file

English version:
Sbornik: Mathematics, 1995, 186:2, 197–219

Bibliographic databases:

Document Type: Article
UDC: 517.9
MSC: Primary 35J25; Secondary 47F05, 47N20
Received: 10.11.1994

Citation: A. K. Gushchin, V. P. Mikhailov, “On the continuity of the solutions of a class of non-local problems for an elliptic equation”, Mat. Sb., 186:2 (1995), 37–58; Sb. Math., 186:2 (1995), 197–219

Citation in format AMSBIB
\Bibitem{GusMik95} \by A.~K.~Gushchin, V.~P.~Mikhailov \paper On the continuity of the~solutions of a~class of non-local problems for an~elliptic equation \jour Mat. Sb. \yr 1995 \vol 186 \issue 2 \pages 37--58 \mathnet{http://mi.mathnet.ru/msb12} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1330589} \zmath{https://zbmath.org/?q=an:0849.35025} \transl \jour Sb. Math. \yr 1995 \vol 186 \issue 2 \pages 197--219 \crossref{https://doi.org/10.1070/SM1995v186n02ABEH000012} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1995RZ91900012} 

Linking options:
• http://mi.mathnet.ru/eng/msb12
• http://mi.mathnet.ru/eng/msb/v186/i2/p37

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. K. Gushchin, “Some properties of the solutions of the Dirichlet problem for a second-order elliptic equation”, Sb. Math., 189:7 (1998), 1009–1045
2. Gushchin, AK, “A condition for complete continuity of the operators arising in nonlocal problems for elliptic equations”, Doklady Mathematics, 62:1 (2000), 32
3. A. K. Gushchin, “A condition for the compactness of operators in a certain class and its application to the analysis of the solubility of non-local problems for elliptic equations”, Sb. Math., 193:5 (2002), 649–668
4. A. K. Gushchin, “A strengthening of the interior Hölder continuity property for solutions of the Dirichlet problem for a second-order elliptic equation”, Theoret. and Math. Phys., 157:3 (2008), 1655–1670
5. Jangveladze T.A., Lobjanidze G.B., “On a variational statement of a nonlocal boundary value problem for a fourth-order ordinary differential equation”, Differ. Equ., 45:3 (2009), 335–343
6. P. L. Gurevich, “Elliptic problems with nonlocal boundary conditions and Feller semigroups”, Journal of Mathematical Sciences, 182:3 (2012), 255–440
7. A. K. Guschin, “Otsenki resheniya zadachi Dirikhle s granichnoi funktsiei iz $L_p$”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 1(22) (2011), 53–67
8. Dzhangveladze T.A., Lobzhanidze G.B., “Ob odnoi nelokalnoi kraevoi zadache dlya obyknovennogo differentsialnogo uravneniya chetvertogo poryadka”, Differentsialnye uravneniya, 47:2 (2011), 181–188
9. Jangveladze T.A., Lobjanidze G.B., “On a Nonlocal Boundary Value Problem for a Fourth-Order Ordinary Differential Equation”, Differ. Equ., 47:2 (2011), 179–186
10. A. K. Gushchin, “The Dirichlet problem for a second-order elliptic equation with an $L_p$ boundary function”, Sb. Math., 203:1 (2012), 1–27
11. V. P. Mikhailov, “O suschestvovanii granichnykh znachenii u reshenii ellipticheskikh uravnenii”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 1(30) (2013), 97–105
12. A. K. Guschin, “$L_p$-otsenki nekasatelnoi maksimalnoi funktsii dlya reshenii ellipticheskogo uravneniya vtorogo poryadka”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 1(30) (2013), 53–69
13. A. K. Gushchin, “$L_p$-estimates for solutions of second-order elliptic equation Dirichlet problem”, Theoret. and Math. Phys., 174:2 (2013), 209–219
14. Temur Jangveladze, Zurab Kiguradze, George Lobjanidze, “Variational Statement and Domain Decomposition Algorithms for Bitsadze-Samarskii Nonlocal Boundary Value Problem for Poisson’s Two-Dimensional Equation”, International Journal of Partial Differential Equations, 2014 (2014), 1
15. A. K. Gushchin, “$L_p$-estimates for the nontangential maximal function of the solution to a second-order elliptic equation”, Sb. Math., 207:10 (2016), 1384–1409
16. A. K. Gushchin, “The Luzin area integral and the nontangential maximal function for solutions to a second-order elliptic equation”, Sb. Math., 209:6 (2018), 823–839
•  Number of views: This page: 260 Full text: 65 References: 19 First page: 3

 Contact us: math-net2019_03 [at] mi-ras ru Terms of Use Registration Logotypes © Steklov Mathematical Institute RAS, 2019