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Mat. Sb. (N.S.), 1988, Volume 137(179), Number 1(9), Pages 19–64 (Mi msb1764)  

This article is cited in 45 scientific papers (total in 45 papers)

On the Dirichlet problem for a second-order elliptic equation

A. K. Gushchin


Abstract: The function space $C_{n-1}(\overline Q)$, $C(\overline Q)\subset C_{n-1}(\overline Q)\subset L_2(Q)$, where $Q$ is a bounded domain in $\mathbf R_n$, consists of elements that on sets of positive $(n-1)$-dimensional Hausdorff measure have traces with a property analogous to joint continuity. For $\partial Q\in C^1$ the set of traces of the functions in $C_{n-1}(\overline Q)$ on $\partial Q$ coincides with $L_2(\partial Q)$, and the imbedding $W_2^1(Q)\subset C_{n-1}(\overline Q)$ is valid.
Solutions of the Dirichlet problem in $C_{n-1}(\overline Q)$ are considered for the elliptic equation
$$ \sum_{i,j=1}^n(a_{ij}(x)u_{x_i})_{x_j}=f,\quad x\in Q;\qquad u|_{\partial Q}=u_0. $$
Under the assumption that the normal to $\partial Q$ and the coefficients of the equation satisfy the Dini condition on $\partial Q$, it is established that for all $u_0\in L_2(\partial Q)$ and $f\in W_2^{-1}(Q)$ there is a unique solution that depends continuously on $u_0$ and $f$. It is proved that in this situation the solution in $C_{n-1}(\overline Q)$ coincides with the concept of a solution in $W^1_{2,\mathrm{loc}}$ introduced by Mikhailov.
Bibliography: 39 titles.

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English version:
Mathematics of the USSR-Sbornik, 1990, 65:1, 19–66

Bibliographic databases:

Document Type: Article
UDC: 517.9
MSC: Primary 35J25, 35D05; Secondary 35B45
Received: 07.12.1987

Citation: A. K. Gushchin, “On the Dirichlet problem for a second-order elliptic equation”, Mat. Sb. (N.S.), 137(179):1(9) (1988), 19–64; Math. USSR-Sb., 65:1 (1990), 19–66

Citation in format AMSBIB
\Bibitem{Gus88}
\by A.~K.~Gushchin
\paper On the Dirichlet problem for a second-order elliptic equation
\jour Mat. Sb. (N.S.)
\yr 1988
\vol 137(179)
\issue 1(9)
\pages 19--64
\mathnet{http://mi.mathnet.ru/msb1764}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=965878}
\zmath{https://zbmath.org/?q=an:0683.35013}
\transl
\jour Math. USSR-Sb.
\yr 1990
\vol 65
\issue 1
\pages 19--66
\crossref{https://doi.org/10.1070/SM1990v065n01ABEH002075}


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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, V. P. Mikhailov, “On the existence of boundary values of solutions of an elliptic equation”, Math. USSR-Sb., 73:1 (1992), 171–194  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    2. Gushchin A., Mikhailov V., “On Fredholmeness of Some Nonlocal Problems for an Elliptic-Equations of the 2nd-Order”, Dokl. Akad. Nauk, 333:3 (1993), 290–292  mathnet  mathscinet  zmath  isi
    3. 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
    4. 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  mathnet  crossref  mathscinet  zmath  isi
    5. Gushchin A., Mikhailov V., “On Unique Solvability of Non-Local Problems for Elliptic Equation”, Dokl. Akad. Nauk, 351:1 (1996), 7–8  mathnet  mathscinet  zmath  isi
    6. 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
    7. Gushchin A., “Inner Estimates of the Weak Solution of the Dirichlet Problem for a Second-Order Elliptic Equation”, Dokl. Akad. Nauk, 358:6 (1998), 731–733  mathnet  mathscinet  zmath  isi
    8. 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
    9. Gushchin, AK, “A condition for complete continuity of the operators arising in nonlocal problems for elliptic equations”, Doklady Mathematics, 62:1 (2000), 32  isi  elib
    10. 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
    11. 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  zmath  isi
    12. V. P. Mikhailov, “Existence of boundary values for biharmonic functions”, Sb. Math., 195:12 (2004), 1781–1793  mathnet  crossref  crossref  mathscinet  zmath  isi
    13. Gushchin, AK, “Carleson-type estimates for solutions to second-order elliptic equations”, Doklady Mathematics, 69:3 (2004), 329  isi  elib
    14. Mikhailov, VP, “On the existence of limit values of a biharmonic function on the boundary of a domain”, Doklady Mathematics, 69:2 (2004), 228  isi  elib
    15. 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
    16. Gushchin, AK, “On the interior smoothness of solutions to second-order elliptic equations”, Doklady Mathematics, 72:2 (2005), 665  isi  elib
    17. 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  isi
    18. 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
    19. 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
    20. 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
    21. 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
    22. V. F. Gilimshina, F. Kh. Mukminov, “On the decay of solutions of non-uniformly elliptic equations”, Izv. Math., 75:1 (2011), 53–71  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    23. 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
    24. 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
    25. Dumanyan V.Zh., “O razreshimosti zadachi dirikhle dlya obschego ellipticheskogo uravneniya vtorogo poryadka”, Doklady Akademii nauk, 436:2 (2011), 159–162  elib
    26. 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
    27. 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
    28. 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
    29. 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
    30. 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
    31. Gushchin A.K., “Estimates of the Nontangential Maximal Function for Solutions of a Second-Order Elliptic Equation”, Dokl. Math., 86:2 (2012), 667–669  mathnet  crossref  isi  elib
    32. 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
    33. 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
    34. 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
    35. V. Zh. Dumanyan, “On boundedness of a class of first order linear differential operators in the space of $(n-1)$-dimensionally continuous functions”, Uch. zapiski EGU, ser. Fizika i Matematika, 2013, no. 2, 8–14  mathnet
    36. 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
    37. 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
    38. 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
    39. 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
    40. 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
    41. A. K. Gushchin, “$L_p$-estimates for the nontangential maximal function of the solution to a second-order elliptic equation”, Sb. Math., 207:10 (2016), 1384–1409  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    42. 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
    43. 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  mathnet  crossref  crossref  adsnasa  isi  elib
    44. 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
    45. 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
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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