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

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

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
\Bibitem{Maz72} \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 \mathnet{http://mi.mathnet.ru/msb3133} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=312057} \zmath{https://zbmath.org/?q=an:0262.35024} \transl \jour Math. USSR-Sb. \yr 1972 \vol 16 \issue 3 \pages 429--469 \crossref{https://doi.org/10.1070/SM1972v016n03ABEH001435} 

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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
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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
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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
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
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
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