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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.

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

Bibliographic databases:

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
\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
\jour Sb. Math.
\yr 1995
\vol 186
\issue 2
\pages 197--219

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    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  mathnet  crossref  crossref  mathscinet  zmath  isi
    2. Gushchin, AK, “A condition for complete continuity of the operators arising in nonlocal problems for elliptic equations”, Doklady Mathematics, 62:1 (2000), 32  mathscinet  zmath  isi  elib
    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  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    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  crossref  mathscinet  zmath  isi  elib
    6. P. L. Gurevich, “Elliptic problems with nonlocal boundary conditions and Feller semigroups”, Journal of Mathematical Sciences, 182:3 (2012), 255–440  mathnet  crossref  mathscinet  zmath
    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  mathnet  crossref  elib
    8. 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  crossref  mathscinet  zmath  isi  elib
    9. 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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    10. 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  mathnet  crossref
    11. 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  mathnet  crossref
    12. A. K. Gushchin, “$L_p$-estimates for solutions of second-order elliptic equation Dirichlet problem”, Theoret. and Math. Phys., 174:2 (2013), 209–219  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    13. 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  crossref  zmath
    14. 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  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    15. 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  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    16. Z. A. Nakhusheva, “Kraevye zadachi s integralnym smescheniem dlya modelnogo uravneniya ellipticheskogo tipa”, Vestnik KRAUNTs. Fiz.-mat. nauki, 32:3 (2020), 65–74  mathnet  crossref
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