On the theory of solvability of a~problem with oblique derivative

B. P. Paneah

Sbornik. Mathematics, Tome 42 (1982) no. 2, pp. 197-235

Résumé

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.