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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.
UDC:
517.946.9
MSC: Primary
35J70; Secondary
35S15 Received: 21.05.1980
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