Analyticity and criticality results for the eigenvalues of the biharmonic operator

TL;DR

This paper studies the eigenvalues of the biharmonic operator under various boundary conditions, proving their analyticity, deriving shape derivatives, and showing that balls are critical domains in shape optimization.

Contribution

It establishes the analyticity of eigenvalues and symmetric functions, derives shape derivatives, and proves that balls are critical domains under volume constraints.

Findings

Eigenvalues are real analytic functions.

Shape derivatives are derived using Hadamard-type formulas.

Balls are critical domains in shape optimization.

Abstract

We consider the eigenvalues of the biharmonic operator subject to several homogeneous boundary conditions (Dirichlet, Neumann, Navier, Steklov). We show that simple eigenvalues and elementary symmetric functions of multiple eigenvalues are real analytic, and provide Hadamard-type formulas for the corresponding shape derivatives. After recalling the known results in shape optimization, we prove that balls are always critical domains under volume constraint.