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

## TL;DR

This paper investigates the completeness of root functions for Sturm-Liouville problems associated with the Lamé system in weighted spaces, establishing conditions for Fredholm properties and root function completeness.

## Contribution

It provides new criteria for the completeness of root functions of Sturm-Liouville problems for the Lamé system in weighted Sobolev spaces, including both coercive and non-coercive cases.

## Key findings

- Problems are shown to be Fredholm in weighted Sobolev spaces.
- Conditions for the completeness of root functions are explicitly described.
- Results apply to boundary value problems with Robin boundary conditions.

## 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.