Minimization of the first eigenvalue for the Lam\'e system

TL;DR

This paper investigates the shape optimization problem for the first eigenvalue of the Lamé system, establishing existence of optimal domains, and analyzing conditions under which the disk is or isn't optimal based on the Poisson ratio.

Contribution

It proves the existence of optimal domains for the Lamé system eigenvalue minimization and characterizes when the disk is a local minimizer depending on the Poisson ratio.

Findings

Optimal domains exist within quasi-open sets.

The disk is not optimal for Poisson ratios below a certain threshold.

The disk becomes a local minimizer when the Poisson ratio exceeds this threshold.

Abstract

In this article, we address the problem of determining a domain in $\mathbb{R}^N$ that minimizes the first eigenvalue of the Lam\'e system under a volume constraint. We begin by establishing the existence of such an optimal domain within the class of quasi-open sets, showing that in the physically relevant dimensions $N = 2$ and $3$, the optimal domain is indeed an open set. Additionally, we derive both first and second-order optimality conditions. Leveraging these conditions, we demonstrate that in two dimensions, the disk cannot be the optimal shape when the Poisson ratio is below a specific threshold, whereas above this value, it serves as a local minimizer. We also extend our analysis to show that the disk is nonoptimal for Poisson ratios $\nu$ satisfying $\nu \leq 0.4$.