Optimization of the Steklov-Lam\'e eigenvalues with respect to the domain

TL;DR

This paper investigates the optimization of Steklov-Lamé eigenvalues on variable domains, establishing bounds, shape maximization results, and proposing a numerical method for eigenvalue computation.

Contribution

It provides new theoretical results on eigenvalue maximization under geometric constraints and introduces a numerical approach for computing these eigenvalues.

Findings

The disk maximizes the first non-zero eigenvalue under area or perimeter constraints.

Eigenvalues are upper semicontinuous under domain convergence.

Numerical methods effectively compute eigenvalues for shape optimization studies.

Abstract

This work deals with theoretical and numerical aspects related to the behavior of the Steklov-Lam\'e eigenvalues on variable domains. After establishing the eigenstructure for the disk, we prove that for a certain class of Lam\'e parameters, the disk maximizes the first non-zero eigenvalue under area or perimeter constraints in dimension two. Upper bounds for these eigenvalues can be found in terms of the scalar Steklov eigenvalues, involving various geometric quantities. We prove that the Steklov-Lam\'e eigenvalues are upper semicontinuous for the complementary Hausdorff convergence of $\varepsilon$-cone domains and, as a consequence, there exist shapes maximizing these eigenvalues under convexity and volume constraints. A numerical method based on fundamental solutions is proposed for computing the Steklov-Lam\'e eigenvalues, allowing to study numerically the shapes maximizing the first ten non-zero eigenvalues.