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This article is cited in 13 scientific papers (total in 13 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 \quad\text{in}\quad\Omega,\qquad
\frac{\partial u}{\partial\mathbf l}+bu=f\quad\text{on}\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 \quad\text{in}\quad\Omega,\qquad
\frac{\partial u}{\partial\mathbf l}+bu=f\quad\text{on}\quad M\setminus\mu_2,
\qquad u=g\quad\text{on}\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.
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
Linking options:
https://www.mathnet.ru/eng/sm2321https://doi.org/10.1070/SM1982v042n02ABEH002251 https://www.mathnet.ru/eng/sm/v156/i2/p226
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Abstract page: | 347 | Russian version PDF: | 126 | English version PDF: | 14 | References: | 46 |
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