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Mat. Sb., 1991, Volume 182, Number 6, Pages 787–810 (Mi msb1324)  

This article is cited in 34 scientific papers (total in 34 papers)

On the existence of boundary values of solutions of an elliptic equation

A. K. Gushchin, V. P. Mikhailov


Abstract: A test is established for the existence of a boundary value for the solution of the elliptic second order equation
$$ -\sum_{i,j=1}^n(a_{ij}(x)u_{x_i}(x))_{x_j}=f(x)-\operatorname{div}F(x), \quad x\in Q. $$
In this connection, it is proved that the solution has a property $(u\in C_{n-1}(\overline Q))$ similar to continuity with respect to all variables in $\overline Q$, and that its boundary value $u|_{\partial Q}\in L_2(\partial Q)$ is the limit in $L_2$ of the traces of the solution on surfaces in a large class (which are not necessarily “parallel” to the boundary).

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English version:
Mathematics of the USSR-Sbornik, 1992, 73:1, 171–194

Bibliographic databases:

UDC: 517.9
MSC: 35J67
Received: 04.12.1990

Citation: A. K. Gushchin, V. P. Mikhailov, “On the existence of boundary values of solutions of an elliptic equation”, Mat. Sb., 182:6 (1991), 787–810; Math. USSR-Sb., 73:1 (1992), 171–194

Citation in format AMSBIB
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\by A.~K.~Gushchin, V.~P.~Mikhailov
\paper On~the existence of boundary values of solutions of an elliptic equation
\jour Mat. Sb.
\yr 1991
\vol 182
\issue 6
\pages 787--810
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\zmath{https://zbmath.org/?q=an:0781.35020}
\adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?1992SbMat..73..171G}
\transl
\jour Math. USSR-Sb.
\yr 1992
\vol 73
\issue 1
\pages 171--194
\crossref{https://doi.org/10.1070/SM1992v073n01ABEH002540}
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Vlasov V., Rachkov A., “On Weighted Hardy-Type Spaces”, Dokl. Akad. Nauk, 328:3 (1993), 281–284  mathnet  mathscinet  zmath  isi
    2. A. K. Gushchin, V. P. Mikhailov, “On solvability of nonlocal problems for a second-order elliptic equation”, Russian Acad. Sci. Sb. Math., 81:1 (1995), 101–136  mathnet  crossref  mathscinet  zmath  isi
    3. 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
    4. 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
    5. 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
    6. 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
    7. Dumanian V., “Near-Boundary Behavior of the Solution to the Dirichlet Problem for an Elliptic Second-Order Equation”, Dokl. Math., 66:2 (2002), 257–259  mathscinet  isi
    8. Gushchin, AK, “Carleson-type estimates for solutions to second-order elliptic equations”, Doklady Mathematics, 69:3 (2004), 329  zmath  isi  elib
    9. 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
    10. Gushchin, AK, “On the interior smoothness of solutions to second-order elliptic equations”, Doklady Mathematics, 72:2 (2005), 665  mathscinet  zmath  isi  elib
    11. 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
    12. 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
    13. V. Zh. Dumanyan, “Dirichlet weight integral estimation to Dirichlet problem solution for the general second order elliptic equations”, Uch. zapiski EGU, ser. Fizika i Matematika, 2009, no. 3, 10–21  mathnet
    14. V. P. Mikhailov, “Existence of boundary values of polyharmonic functions”, Sb. Math., 201:5 (2010), 735–757  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    15. Dumanyan V.Zh., “On Boundary Values of Solutions of the Dirichlet Problem for Second Order Elliptic Equation”, J. Contemp. Math. Anal.-Armen. Aca., 45:1 (2010), 26–42  crossref  mathscinet  isi
    16. 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
    17. V. Zh. Dumanyan, “Solvability of the Dirichlet problem for a general second-order elliptic equation”, Sb. Math., 202:7 (2011), 1001–1020  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    18. Dumanyan V.Zh., “O razreshimosti zadachi dirikhle dlya obschego ellipticheskogo uravneniya vtorogo poryadka”, Doklady Akademii nauk, 436:2 (2011), 159–162  elib
    19. Guschin A.K., “O razreshimosti zadachi dirikhle s granichnoi funktsiei iz l_{p} dlya ellipticheskogo uravneniya vtorogo poryadka”, Doklady Akademii nauk, 437:5 (2011), 583–586  elib
    20. Gushchin A.K., “Solvability of the Dirichlet Problem for a Second-Order Elliptic Equation with a Boundary Function From l-P”, Dokl. Math., 83:2 (2011), 219–221  crossref  mathscinet  zmath  isi  elib
    21. Dumanyan V.Zh., “Solvability of the Dirichlet Problem for the General Second-Order Elliptic Equation”, Dokl. Math., 83:1 (2011), 30–33  crossref  mathscinet  zmath  isi
    22. Dumanyan V.Zh., “On Solvability of Dirichlet Problem for Second Order Elliptic Equation”, J. Contemp. Math. Anal.-Armen. Aca., 46:2 (2011), 77–88  crossref  mathscinet  isi
    23. 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
    24. 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
    25. V. Zh. Dumanyan, “Solvability of the Dirichlet problem for second-order elliptic equations”, Theoret. and Math. Phys., 180:2 (2014), 917–931  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
    26. A. K. Guschin, “O zadache Dirikhle dlya ellipticheskogo uravneniya”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 19:1 (2015), 19–43  mathnet  crossref  zmath  elib
    27. A. K. Gushchin, “V.A. Steklov's work on equations of mathematical physics and development of his results in this field”, Proc. Steklov Inst. Math., 289 (2015), 134–151  mathnet  crossref  crossref  isi  elib
    28. A. K. Gushchin, “Solvability of the Dirichlet problem for an inhomogeneous second-order elliptic equation”, Sb. Math., 206:10 (2015), 1410–1439  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    29. Dumanyan V.Zh., “on Solvability of the Dirichlet Problem With the Boundary Function in l (2) For a Second-Order Elliptic Equation”, J. Contemp. Math. Anal.-Armen. Aca., 50:4 (2015), 153–166  crossref  mathscinet  isi
    30. Petrushko I.M., “On Boundary and Initial Values of Solutions of a Second-Order Parabolic Equation That Degenerate on the Domain Boundary”, Dokl. Math., 96:3 (2017), 568–570  crossref  isi
    31. A. K. Gushchin, “A criterion for the existence of $L_p$ boundary values of solutions to an elliptic equation”, Proc. Steklov Inst. Math., 301 (2018), 44–64  mathnet  crossref  crossref  isi  elib  elib
    32. N. A. Gusev, “On the definitions of boundary values of generalized solutions to an elliptic-type equation”, Proc. Steklov Inst. Math., 301 (2018), 39–43  mathnet  crossref  crossref  isi  elib  elib
    33. Vlasov I V., “Hardy Spaces, Approximation Issues and Boundary Value Problems”, Eurasian Math. J., 9:3 (2018), 85–94  mathnet  crossref  mathscinet  isi  scopus
    34. 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 - 1991 Sbornik: Mathematics (from 1967)
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