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Mat. Sb., 2002, Volume 193, Number 5, Pages 17–36 (Mi msb649)  

This article is cited in 17 scientific papers (total in 17 papers)

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

A. K. Gushchin

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: A class of “integral” operators arising in the analysis of non-local problems in which the values of a solution at the boundary of the domain under consideration are expressed through its values at interior points is investigated. These operators are defined in terms of measures close to Carleson measures. A condition ensuring the complete continuity of such operators is found. This result enables one to complement and extend results on the Fredholm property of a broad class of non-local problems for a second-order elliptic equation.

DOI: https://doi.org/10.4213/sm649

Full text: PDF file (352 kB)
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English version:
Sbornik: Mathematics, 2002, 193:5, 649–668

Bibliographic databases:

UDC: 517.9
MSC: Primary 35J35; Secondary 47G10
Received: 28.12.2001

Citation: 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”, Mat. Sb., 193:5 (2002), 17–36; Sb. Math., 193:5 (2002), 649–668

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. A. K. Gushchin, “Carleson-type estimates for solutions to second-order elliptic equations”, Dokl. Math., 69:3 (2004), 329–331  mathnet  mathscinet  zmath  isi  elib
    2. P. L. Gurevich, “Generalized Solutions of Nonlocal Elliptic Problems”, Math. Notes, 77:5 (2005), 614–629  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    3. 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
    4. A. K. Gushchin, “On the interior smoothness of solutions to second-order elliptic equations”, Dokl. Math., 72:2 (2005), 665–668  mathnet  mathscinet  mathscinet  zmath  isi  elib  elib
    5. P. L. Gurevich, “On the Stability of the Index of Unbounded Nonlocal Operators in Sobolev Spaces”, Proc. Steklov Inst. Math., 255 (2006), 108–126  mathnet  crossref  mathscinet  elib
    6. A. L. Skubachevskii, “Nonclassical boundary value problems. I”, Journal of Mathematical Sciences, 155:2 (2008), 199–334  mathnet  crossref  mathscinet  zmath  elib
    7. A. K. Gushchin, “Smoothness of solutions to the Dirichlet problem for a second-order elliptic equation with a square integrable boundary function”, Dokl. Math., 76:1 (2007), 486–489  mathnet  crossref  mathscinet  zmath  isi  elib  elib  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. A. L. Skubachevskii, “Nonclassical boundary-value problems. II”, Journal of Mathematical Sciences, 166:4 (2010), 377–561  mathnet  crossref  mathscinet  elib
    10. N. V. Beilina, “O suschestvovanii resheniya odnoi nelokalnoi zadachi dlya uravneniya Laplasa”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 1(20) (2010), 205–208  mathnet  crossref
    11. 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
    12. 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
    13. 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
    14. 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
    15. 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
    16. A. I. Kozhanov, G. A. Lukina, “Nelokalnye zadachi s integralnym usloviem dlya differentsialnykh uravnenii nechetnogo poryadka”, Sib. elektron. matem. izv., 13 (2016), 452–466  mathnet  crossref  mathscinet
    17. Kozhanov A.I., “Nonlocal Problems With Integral Conditions For Elliptic Equations”, Complex Var. Elliptic Equ., 64:5 (2019), 741–752  crossref  mathscinet  zmath  isi  scopus
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