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 Mat. Sb., 1998, Volume 189, Number 7, Pages 53–90 (Mi msb337)

This article is cited in 13 scientific papers (total in 13 papers)

Some properties of the solutions of the Dirichlet problem for a second-order elliptic equation

A. K. Gushchin

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: The paper is devoted to the identification of the properties of the solution of the Dirichlet problem for a second-order elliptic equation with boundary function in $L_2$ that characterize its behaviour near the boundary of the domain under consideration. In particular, we study the behaviour of the integrals of the derivatives of the solution with respect to measures concentrated to a considerable extent on sets of various dimensions approaching the boundary. The corresponding description is given in terms of special function spaces that reflect the interior regularity of the solution and some of its integral properties. The results obtained are applied to the study of the Fredholm property for a wide class of non-local problems, in which the boundary values of a solution are related to its values and the values of its derivatives at interior points.

DOI: https://doi.org/10.4213/sm337

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English version:
Sbornik: Mathematics, 1998, 189:7, 1009–1045

Bibliographic databases:

Document Type: Article
UDC: 517.9
MSC: Primary 35J25; Secondary 46E15
Received: 25.12.1997

Citation: A. K. Gushchin, “Some properties of the solutions of the Dirichlet problem for a second-order elliptic equation”, Mat. Sb., 189:7 (1998), 53–90; Sb. Math., 189:7 (1998), 1009–1045

Citation in format AMSBIB
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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. I. M. Petrushko, “Existence of boundary values for solutions of degenerate elliptic equations”, Sb. Math., 190:7 (1999), 973–1004
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. Gushchin, AK, “Carleson-type estimates for solutions to second-order elliptic equations”, Doklady Mathematics, 69:3 (2004), 329
5. A. K. Gushchin, “On the interior smoothness of solutions to second-order elliptic equations”, Siberian Math. J., 46:5 (2005), 826–840
6. Gushchin, AK, “On the interior smoothness of solutions to second-order elliptic equations”, Doklady Mathematics, 72:2 (2005), 665
7. Gushchin, AK, “Smoothness of solutions to the Dirichlet problem for a second-order elliptic equation with a square integrable boundary function”, Doklady Mathematics, 76:1 (2007), 486
8. 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
9. L. M. Kozhevnikova, “Behaviour at infinity of solutions of pseudodifferential elliptic equations in unbounded domains”, Sb. Math., 199:8 (2008), 1169–1200
10. A. R. Gerfanov, F. Kh. Mukminov, “Shirokii klass edinstvennosti resheniya dlya neravnomerno ellipticheskogo uravneniya v neogranichennoi oblasti”, Ufimsk. matem. zhurn., 1:3 (2009), 11–27
11. 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
12. 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
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
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