Eigenvalues of poly-harmonic operators on variable domains

TL;DR

This paper studies how eigenvalues of poly-harmonic operators change with domain shape, providing formulas for their sensitivity, identifying critical shapes like balls, and analyzing domain perturbations.

Contribution

It introduces new analyticity results, Hadamard-type formulas, and characterizations of critical domains for eigenvalues of poly-harmonic operators.

Findings

Eigenvalues depend analytically on domain perturbations.

Balls are identified as critical domains for eigenvalues.

Formulas for the derivatives of eigenvalues with respect to domain changes.

Abstract

We consider a class of eigenvalue problems for poly-harmonic operators, including Dirichlet and buckling-type eigenvalue problems. We prove an analyticity result for the dependence of the symmetric functions of the eigenvalues upon domain perturbations and compute Hadamard-type formulas for the Frech\'{e}t differentials. We also consider isovolumetric domain perturbations and characterize the corresponding critical domains for the symmetric functions of the eigenvalues. Finally, we prove that balls are critical domains.