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Mat. Sb. (N.S.), 1972, Volume 87(129), Number 3, Pages 417–454 (Mi msb3133)  

This article is cited in 27 scientific papers (total in 27 papers)

On a degenerating problem with directional derivative

V. G. Maz'ya

Abstract: We study a problem with directional derivative for a second order elliptic equation. We assume that smooth compact submanifolds $\Gamma_0\supset\Gamma_1\supset\cdots\supset\Gamma_s$ have been selected from the boundary $\Gamma$, and that the vector field is tangent to $\Gamma_i$ ($i\leqslant s-1$) at points of $\Gamma_{i+1}$ and not tangent to $\Gamma_s$. We show that the problem has a unique solution, obtain estimates of the solutions in $L_p(\Gamma)$ ($1<p\leqslant\infty$), and prove that the inverse operator is compact.
Bibliography: 29 titles.

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English version:
Mathematics of the USSR-Sbornik, 1972, 16:3, 429–469

Bibliographic databases:

UDC: 517.9
MSC: Primary 35J70; Secondary 35J25, 35S15
Received: 29.03.1971

Citation: V. G. Maz'ya, “On a degenerating problem with directional derivative”, Mat. Sb. (N.S.), 87(129):3 (1972), 417–454; Math. USSR-Sb., 16:3 (1972), 429–469

Citation in format AMSBIB
\by V.~G.~Maz'ya
\paper On a~degenerating problem with directional derivative
\jour Mat. Sb. (N.S.)
\yr 1972
\vol 87(129)
\issue 3
\pages 417--454
\jour Math. USSR-Sb.
\yr 1972
\vol 16
\issue 3
\pages 429--469

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    This publication is cited in the following articles:
    1. Bengt Winzell, “The oblique derivative problem II”, Ark Mat, 17:1-2 (1979), 107  crossref  mathscinet  zmath
    2. B. P. Paneah, “On the theory of solvability of a problem with oblique derivative”, Math. USSR-Sb., 42:2 (1982), 197–235  mathnet  crossref  mathscinet  zmath
    3. Bengt Winzell, “A boundary value problem with an oblique derivative”, Communications in Partial Differential Equations, 6:3 (1981), 305  crossref
    4. 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
    5. B. P. Paneah, “Some boundary value problems for elliptic equations, and the Lie algebras associated with them”, Math. USSR-Sb., 54:1 (1986), 207–237  mathnet  crossref  mathscinet  zmath
    6. Panejah B., “Nonelliptic Boundary-Problems and Related Lie-Algebras”, 283, no. 1, 1985, 49–53  isi
    7. 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
    8. B. P. Paneah, “Some boundary value problems for elliptic equations, and the Lie algebras connected with them. II”, Math. USSR-Sb., 61:2 (1988), 495–527  mathnet  crossref  mathscinet  zmath
    9. Alimov S., “Smoothness of a Solution of a Degenerate Problem Involving a Directional Derivative”, Differ. Equ., 23:1 (1987), 1–10  isi
    10. 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
    11. 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
    12. V. G. Maz'ya, T. O. Shaposhnikova, “Sharp Pointwise Interpolation Inequalities for Derivatives”, Funct. Anal. Appl., 36:1 (2002), 30–48  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    13. P. Popivanov, N. Kutev, “Viscosity solutions to the degenerate oblique derivative problem for fully nonlinear elliptic equations”, Math Nachr, 278:7-8 (2005), 888  crossref  mathscinet  zmath  isi
    14. L.G. Softova, “-Solvability for the Parabolic Poincaré Problem”, Communications in Partial Differential Equations, 29:11-12 (2005), 1783  crossref
    15. Dian K. Palagachev, “The Poincaré problem in -Sobolev spaces—I: codimension one degeneracy”, Journal of Functional Analysis, 229:1 (2005), 121  crossref
    16. Dian K. Palagachev, “Neutral Poincaré problem in Lp-Sobolev spaces: Regularity and Fredholmness”, Internat Math Res Notices, 2006 (2006), 1  crossref
    17. Dian K. Palagachev, “ W 2,p -a priori estimates for the emergent Poincaré Problem”, J Global Optim, 40:1-3 (2008), 305  crossref  mathscinet  zmath  isi
    18. Dian K. Palagachev, “The Poincaré Problem inLp-Sobolev Spaces II: Full Dimension Degeneracy”, Communications in Partial Differential Equations, 33:2 (2008), 209  crossref
    19. 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
    20. 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
    21. 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  isi
    22. 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
    23. Burskii V.P. Lesina E.V., “Neumann Problem and One Oblique-Derivative Problem for an Improperly Elliptic Equation”, Ukr. Math. J., 64:4 (2012), 511–524  crossref  isi
    24. 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
    25. 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
    26. 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
    27. V. I. Vlasov, “Hardy spaces, approximation issues and boundary value problems”, Eurasian Math. J., 9:3 (2018), 85–94  mathnet  crossref
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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