Sturm--Liouville problems in weighted spaces in domains with nonsmooth edges.~II

N. Tarkhanov ; A. A. Shlapunov

Matematičeskie trudy, Tome 18 (2015) no. 2, pp. 133-204

Résumé

We consider a (generally, noncoercive) mixed boundary value problem in a bounded domain $\mathcal{D}$ of ${\mathbb{R}}^n$ for a second order elliptic differential operator $A (x,\partial)$. The differential operator is assumed to be of divergent form in $\mathcal{D}$ and the boundary operator $B (x,\partial)$ is of Robin type on $\partial \mathcal{D}$. The boundary of $\mathcal{D}$ is assumed to be a Lipschitz surface. Besides, we distinguish a closed subset $Y \subset \partial \mathcal{D}$ and control the growth of solutions near $Y$. We prove that the pair $(A,B)$ induces a Fredholm operator $L$ in suitable weighted spaces of Sobolev type, the weight function being a power of the distance to the singular set $Y$. Moreover, we prove the completeness of root functions related to $L$.