On a classical spectral optimization problem in linear elasticity

TL;DR

This paper surveys classical shape optimization problems related to eigenvalues of elliptic operators, focusing on domain perturbations, critical points, and the role of symmetric domains like balls across various boundary conditions and operators.

Contribution

It provides a comprehensive overview of recent results on the analytic dependence of eigenvalue functions on domain shape and the significance of balls as critical points under volume constraints.

Findings

Balls are critical points for eigenvalue functions under volume constraints.

Eigenvalues depend analytically on domain perturbations.

Various boundary conditions and operators are considered in the analysis.

Abstract

We consider a classical shape optimization problem for the eigenvalues of elliptic operators with homogeneous boundary conditions on domains in the $N$-dimensional Euclidean space. We survey recent results concerning the analytic dependence of the elementary symmetric functions of the eigenvalues upon domain perturbation and the role of balls as critical points of such functions subject to volume constraint. Our discussion concerns Dirichlet and buckling-type problems for polyharmonic operators, the Neumann and the intermediate problems for the biharmonic operator, the Lam\'{e} and the Reissner-Mindlin systems.