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

This article is cited in 10 scientific papers (total in 10 papers)

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
\begin{equation} \mathscr Lu=F \quadin\quad\Omega,\qquad \frac{\partial u}{\partial\mathbf l}+bu=f\quadon\quad M, \end{equation}
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
\begin{equation} \mathscr Lu=F \quadin\quad\Omega,\qquad \frac{\partial u}{\partial\mathbf l}+bu=f\quadon\quad M\setminus\mu_2, \qquad u=g\quadon\quad\mu_1 \end{equation}
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
Received: 21.05.1980

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
\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
\jour Math. USSR-Sb.
\yr 1982
\vol 42
\issue 2
\pages 197--235

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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  isi
    2. Panejah B., “Nonelliptic Boundary-Value-Problems Connected with Diffusion-Processes”, 276, no. 3, 1984, 551–554  mathscinet  isi
    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  mathnet  crossref  mathscinet  zmath
    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  mathnet  crossref  mathscinet  zmath
    5. Alimov S., “Smoothness of a Solution of a Degenerate Problem Involving a Directional Derivative”, Differ. Equ., 23:1 (1987), 1–10  mathscinet  zmath  isi
    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  mathnet  mathscinet  zmath  isi
    7. Dian K. Palagachev, “The Poincaré problem in -Sobolev spaces—I: codimension one degeneracy”, Journal of Functional Analysis, 229:1 (2005), 121  crossref
    8. Dian K. Palagachev, “Neutral Poincaré problem in Lp-Sobolev spaces: Regularity and Fredholmness”, Internat Math Res Notices, 2006 (2006), 1  crossref
    9. Dian K. Palagachev, “The Poincaré Problem inLp-Sobolev Spaces II: Full Dimension Degeneracy”, Communications in Partial Differential Equations, 33:2 (2008), 209  crossref
    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  crossref  mathscinet  zmath  isi
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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