On the completeness of the root functions of the Sturm-Liouville problems for the Lam\'e system in weighted spaces

Abstract

We consider three Sturm--Liouville boundary value problems (the coercive ones and the non-coercive one) in a bounded Lipschitz domain for the perturbed Lam\'e operator with the boundary conditions of Robin type. We prove that the problems are Fredholm ones in proper weighted Sobolev type spaces. The conditions, providing the completeness of the root functions related to the boundary value problem, are described.