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Mat. Sb. (N.S.), 1979, Volume 108(150), Number 1, Pages 3–21 (Mi msb2193)  

This article is cited in 32 scientific papers (total in 32 papers)

On boundary values in $L_p$, $p>1$, of solutions of elliptic equations

A. K. Gushchin, V. P. Mikhailov


Abstract: The behavior near the boundary of generalized solutions of a second order elliptic equation
$$ \sum_{i,j=1}^n\frac\partial{\partial x_i}(a_{ij}(x)\frac{\partial u}{\partial x_j})=f,\qquad x\in Q=\{|x|<1\}\subset\mathbf R_n. $$
in $W_p^1(Q)$, $p>1$, is studied.
It is shown that under a certain condition on the right side of the equation, the boundedness of the function $\|x\|_{L_p(\|x\|=r)}$, $\frac12\leqslant r<1$, is necessary and sufficient for the existence of a limit for the solution $u(rw)$, $\frac12\leqslant r<1$, $|w|=1$, in $L_p(\|w\|=1)$ as $r\to1-0$. Moreover, the summability of the function $(1-|x|)|u(x)|^{p-2}|\nabla u(x)|^2$ is also a necessary and sufficient condition for the existence of a limit in $ L_p$ on the boundary.
Bibliography: 10 titles.

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English version:
Mathematics of the USSR-Sbornik, 1980, 36:1, 1–19

Bibliographic databases:

UDC: 517.9
MSC: Primary 35J67; Secondary 35J25
Received: 07.08.1978

Citation: A. K. Gushchin, V. P. Mikhailov, “On boundary values in $L_p$, $p>1$, of solutions of elliptic equations”, Mat. Sb. (N.S.), 108(150):1 (1979), 3–21; Math. USSR-Sb., 36:1 (1980), 1–19

Citation in format AMSBIB
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\by A.~K.~Gushchin, V.~P.~Mikhailov
\paper On boundary values in $L_p$, $p>1$, of solutions of elliptic equations
\jour Mat. Sb. (N.S.)
\yr 1979
\vol 108(150)
\issue 1
\pages 3--21
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\mathscinet{http://www.ams.org/mathscinet-getitem?mr=524209}
\zmath{https://zbmath.org/?q=an:0453.35035|0434.35032}
\transl
\jour Math. USSR-Sb.
\yr 1980
\vol 36
\issue 1
\pages 1--19
\crossref{https://doi.org/10.1070/SM1980v036n01ABEH001751}
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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. Petrushko I., “On Limiting Values in Lpp Less-Than 1 of Parabolic Equation Solutions”, 251, no. 5, 1980, 1067–1069  mathscinet  zmath  isi
    2. I. M. Petrushko, “On boundary values of solutions of elliptic equations in domains with Lyapunov boundary”, Math. USSR-Sb., 47:1 (1984), 43–72  mathnet  crossref  mathscinet  zmath
    3. Petrushko I., “On Boundary-Order and Initial Values of 2nd-Order Parabolic Equation Solutions in Lp,P Greater-Than 1”, 267, no. 5, 1982, 1063–1066  mathscinet  zmath  isi
    4. Petrushko I., “On the Boundary and Initial Values of the 2nd-Order Parabolic Equation Solutions”, 266, no. 3, 1982, 557–560  mathscinet  zmath  isi
    5. I. M. Petrushko, “On boundary values in $\mathscr L_p$, $p>1$, of solutions of elliptic equations in domains with a Lyapunov boundary”, Math. USSR-Sb., 48:2 (1984), 565–585  mathnet  crossref  mathscinet  zmath
    6. Mikhailov YA., “Boundary-Values in Lp, P-Greater-Than-1, of Solutions of 2nd-Order Linear Elliptic-Equations”, Differ. Equ., 19:2 (1983), 243–258  isi
    7. Fedorova L., “Boundary-Values of Solutions for Inhomogeneous Differentially-Operator Equations”, no. 7, 1983, 21–24  mathscinet  isi
    8. I. M. Petrushko, “On boundary and initial conditions in $\mathscr L_p$, $p>1$, of solutions of parabolic equations”, Math. USSR-Sb., 53:2 (1986), 489–522  mathnet  crossref  mathscinet  zmath
    9. V. Yu. Shelepov, “On boundary properties of solutions of elliptic equations in multidimensional domains representable by means of the difference of convex functions”, Math. USSR-Sb., 61:2 (1988), 437–460  mathnet  crossref  mathscinet  zmath  isi
    10. I. M. Petrushko, “On boundary values of solutions of elliptic equations degenerating on the boundary”, Math. USSR-Sb., 64:1 (1989), 243–262  mathnet  crossref  mathscinet  zmath
    11. A. K. Gushchin, “On the Dirichlet problem for a second-order elliptic equation”, Math. USSR-Sb., 65:1 (1990), 19–66  mathnet  crossref  mathscinet  zmath
    12. Gushchin A., “On the Dirichlet Problem for the 2nd-Order Elliptic Equation”, 302, no. 5, 1988, 1044–1048  mathscinet  isi
    13. V. I. Gorbachuk, A. V. Knyazyuk, “Boundary values of solutions of operator-differential equations”, Russian Math. Surveys, 44:3 (1989), 67–111  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    14. Shelepov V., Tedeyev A., “On One Inequality for Elliptic Equation Solutions and its Applicability in the Theory of Boundary Properties”, 315, no. 1, 1990, 40–43  zmath  isi
    15. 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
    16. L. A. Muravei, A. V. Filinovskii, “On a problem with nonlocal boundary condition for a parabolic equation”, Math. USSR-Sb., 74:1 (1993), 219–249  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    17. Chabrowski J., “The Dirichlet Problem with l(2)-Boundary Data for Elliptic Linear-Equations”, Lect. Notes Math., 1482 (1991), 1–171  crossref  mathscinet  isi
    18. 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
    19. Kapanadze D., “On the Representation of a Harmonic Function by a Simple Layer Potential”, Differ. Equ., 38:2 (2002), 259–262  mathnet  crossref  mathscinet  zmath  isi
    20. 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
    21. 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
    22. 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
    23. 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
    24. 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
    25. 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
    26. 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
    27. 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
    28. 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
    29. S. I. Bezrodnykh, V. I. Vlasov, “Application of the multipole method to direct and inverse problems for the Grad–Shafranov equation with a nonlocal condition”, Comput. Math. Math. Phys., 54:4 (2014), 631–695  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    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. 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, eds. Pasheva V., Popivanov N., Venkov G., Amer Inst Physics, 2018, 040006  crossref  isi
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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