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.
Citation:
A. K. Gushchin, V. P. Mikhailov, “On the continuity of the solutions of a class of non-local problems for an elliptic equation”, Sb. Math., 186:2 (1995), 197–219
\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 Sb. Math.
\yr 1995
\vol 186
\issue 2
\pages 197--219
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\crossref{https://doi.org/10.1070/SM1995v186n02ABEH000012}
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\zmath{https://zbmath.org/?q=an:0849.35025}
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Linking options:
https://www.mathnet.ru/eng/sm12
https://doi.org/10.1070/SM1995v186n02ABEH000012
https://www.mathnet.ru/eng/sm/v186/i2/p37
This publication is cited in the following 16 articles:
Z. A. Nakhusheva, “Kraevye zadachi s integralnym smescheniem dlya modelnogo uravneniya ellipticheskogo tipa”, Vestnik KRAUNTs. Fiz.-mat. nauki, 32:3 (2020), 65–74
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
A. K. Gushchin, “Lp-estimates for the nontangential maximal function of the solution to a second-order elliptic equation”, Sb. Math., 207:10 (2016), 1384–1409
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
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
A. K. Gushchin, “Lp-estimates for solutions of second-order elliptic equation Dirichlet problem”, Theoret. and Math. Phys., 174:2 (2013), 209–219
A. K. Gushchin, “The Dirichlet problem for a second-order elliptic equation with an Lp boundary function”, Sb. Math., 203:1 (2012), 1–27
A. K. Guschin, “Otsenki resheniya zadachi Dirikhle s granichnoi funktsiei iz Lp”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 1(22) (2011), 53–67
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
P. L. Gurevich, “Elliptic problems with nonlocal boundary conditions and Feller semigroups”, Journal of Mathematical Sciences, 182:3 (2012), 255–440
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
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
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
Gushchin, AK, “A condition for complete continuity of the operators arising in nonlocal problems for elliptic equations”, Doklady Mathematics, 62:1 (2000), 32
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
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