Some boundary value problems for elliptic equations, and the Lie algebras connected with them.~II

B. P. Paneah

Sbornik. Mathematics, Tome 61 (1988) no. 2, pp. 495-527

Résumé

This article presents detailed results on solvability and regularity of solutions for the noncoercive boundary value problem $lu=f$ in $\Omega$, $Au=g$ on $\partial\Omega$, where $l$ is a second-order elliptic operator in a bounded region $\Omega\subset\mathbf R^{n+1}$, and $A$ is a second-order operator for which the Lopatinskii conditions are violated on a sufficiently arbitrary subset of $\partial\Omega$. In particular, the principal part of $A$ need not be of definite sign on $T^*(\partial\Omega)$, and this leads (with a view to obtaining well-posed formulations) to the additional condition $u=h$ on $\mu_1$ and to the allowance of a finite discontinuity of $u|_{\partial\Omega}$ on $\mu_2$, where $\mu_1$ and $\mu_2$ are submanifolds of $\partial\Omega$ of codimension 1. The paper encompasses a large part of the known results on the degenerate oblique derivative problem. Bibliography: 10 titles.