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 Mat. Sb. (N.S.), 1981, Volume 114(156), Number 2, Pages 226–268 (Mi msb2321)

On the theory of solvability of a problem with oblique derivative

B. P. Paneah

Abstract: A boundary-value problem with oblique derivative is studied for an elliptic differential operator $\mathscr L=a_{ij}\mathscr D_i\mathscr D_j+a_j\mathscr D_j+a_0$ in a bounded domain $\Omega\in\mathbf R^{n+2}$ with a smooth boundary $M$. It is assumed that the set $\mu$ of those points of $M$ at which the problem's vector field $\mathbf l$ is intersected by the tangent space $T(M)$ is not empty. This is equivalent to the nonellipticity of the boundary-value problem
which can have an infinite-dimensional kernel and cokernel, depending upon the organization of $\mu$ and the behavior of field $\mathbf l$ in a neighborhood of $\mu$. On the set $\mu$, which is permitted to contain a subset of (complete) dimension $n+1$, there are picked out submanifolds $\mu_1$ and $\mu_2$ of codimension 1, transversal to $\mathbf l$, and the problem
is analyzed instead of (1). It is proved that in suitable spaces the operator corresponding to problem (2) is a Fredholm operator and under natural constraints on coefficient $b$ the problem is uniquely solvable in the class of functions $u$ smooth in $[\Omega]\setminus\mu_2$, with a finite jump in $u|_M$. A necessary and sufficient condition is derived for the compactness of the inverse operator of problem (2) in terms of the set $\mu$ and the field $\mathbf l$.
Bibliography: 14 titles.

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English version:
Mathematics of the USSR-Sbornik, 1982, 42:2, 197–235

Bibliographic databases:

UDC: 517.946.9
MSC: Primary 35J70; Secondary 35S15

Citation: B. P. Paneah, “On the theory of solvability of a problem with oblique derivative”, Mat. Sb. (N.S.), 114(156):2 (1981), 226–268; Math. USSR-Sb., 42:2 (1982), 197–235

Citation in format AMSBIB
\Bibitem{Pan81} \by B.~P.~Paneah \paper On the theory of solvability of a~problem with oblique derivative \jour Mat. Sb. (N.S.) \yr 1981 \vol 114(156) \issue 2 \pages 226--268 \mathnet{http://mi.mathnet.ru/msb2321} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=609290} \zmath{https://zbmath.org/?q=an:0484.35031|0457.35028} \transl \jour Math. USSR-Sb. \yr 1982 \vol 42 \issue 2 \pages 197--235 \crossref{https://doi.org/10.1070/SM1982v042n02ABEH002251} 

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This publication is cited in the following articles:
1. Gordin V., Resnyansky Y., “Numerical-Solution of a Problem of Large-Scale Wind-Driven Circulation in the Ocean - a Problem with the Oblique Derivative”, Okeanologiya, 21:6 (1981), 960–965
2. Panejah B., “Nonelliptic Boundary-Value-Problems Connected with Diffusion-Processes”, 276, no. 3, 1984, 551–554
3. 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
4. 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
5. Alimov S., “Smoothness of a Solution of a Degenerate Problem Involving a Directional Derivative”, Differ. Equ., 23:1 (1987), 1–10
6. Pastukhova S., “A Well-Posed Statement of a Mixed Problem with Oblique Derivative for the Wave Operator”, Differ. Equ., 29:8 (1993), 1227–1234
7. Dian K. Palagachev, “The Poincaré problem in -Sobolev spaces—I: codimension one degeneracy”, Journal of Functional Analysis, 229:1 (2005), 121
8. Dian K. Palagachev, “Neutral Poincaré problem in Lp-Sobolev spaces: Regularity and Fredholmness”, Internat Math Res Notices, 2006 (2006), 1
9. Dian K. Palagachev, “The Poincaré Problem inLp-Sobolev Spaces II: Full Dimension Degeneracy”, Communications in Partial Differential Equations, 33:2 (2008), 209
10. 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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