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

This article is cited in 14 scientific papers (total in 14 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.


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

Bibliographic databases:

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
\by A.~K.~Gushchin
\paper Some properties of the~solutions of the~Dirichlet problem for a~second-order elliptic equation
\jour Mat. Sb.
\yr 1998
\vol 189
\issue 7
\pages 53--90
\jour Sb. Math.
\yr 1998
\vol 189
\issue 7
\pages 1009--1045

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    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  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. Gushchin, AK, “Carleson-type estimates for solutions to second-order elliptic equations”, Doklady Mathematics, 69:3 (2004), 329  mathscinet  zmath  isi  elib
    5. A. K. Gushchin, “On the interior smoothness of solutions to second-order elliptic equations”, Siberian Math. J., 46:5 (2005), 826–840  mathnet  crossref  mathscinet  zmath  isi  elib
    6. Gushchin, AK, “On the interior smoothness of solutions to second-order elliptic equations”, Doklady Mathematics, 72:2 (2005), 665  mathscinet  zmath  isi  elib
    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  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    9. L. M. Kozhevnikova, “Behaviour at infinity of solutions of pseudodifferential elliptic equations in unbounded domains”, Sb. Math., 199:8 (2008), 1169–1200  mathnet  crossref  crossref  mathscinet  isi  elib
    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  mathnet  zmath  elib
    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  mathnet  crossref  elib
    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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    14. Petrushko I.M. Petrushko M.I., “On the First Mixed Problem in l-P, P > 1, For the Degenerating on the Boundary Parabolic Equations of Second Order”, AIP Conference Proceedings, 2048, ed. Pasheva V. Popivanov N. Venkov G., Amer Inst Physics, 2018, 040006  crossref  isi
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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