Steklov eigenvalues for the Lam\'e operator in linear elasticity

TL;DR

This paper investigates Steklov eigenvalues associated with the Lamé operator in linear elasticity, establishing spectral properties, numerical approximation methods, and supporting the theory with finite element experiments.

Contribution

It extends Korn's inequality for the eigenproblem and demonstrates convergence of a Galerkin scheme for approximating eigenvalues.

Findings

Existence of a countable spectrum established.

Convergence of the Galerkin scheme proven.

Numerical experiments confirm theoretical results.

Abstract

In this paper we study Steklov eigenvalues for the Lam\'e operator which arise in the theory of linear elasticity. In this eigenproblem the spectral parameter appears in a Robin boundary condition, linking the traction and the displacement. To establish the existence of a countable spectrum for this problem, we present an extension of Korn's inequality. We also show that a proposed conforming Galerkin scheme provides convergent approximations to the true eigenvalues. A standard finite element method is used to conduct numerical experiments on 2D and 3D domains to support our theoretical findings.