Hadamard Asymptotics for Eigenvalues of the Dirichlet Laplacian

TL;DR

This paper investigates the validity of Hadamard's asymptotic formula for Dirichlet Laplacian eigenvalues under boundary perturbations, establishing its applicability for smooth domains and identifying limitations for Lipschitz boundaries.

Contribution

It proves the Hadamard formula holds for $C^1$-domains with $C^1$-perturbations and provides an optimal estimate for the remainder in the $C^{1,eta}$ case, also showing failure for Lipschitz boundaries.

Findings

Hadamard formula valid for $C^1$-domains with $C^1$-perturbations

Optimal estimate derived for $C^{1,eta}$-case

Formula invalid for Lipschitz boundaries

Abstract

This paper is dedicated to the classical Hadamard formula for asymptotics of eigenvalues of the Dirichlet-Laplacian under perturbations of the boundary. We prove that the Hadamard formula still holds for $C^1$-domains with $C^1$-perturbations. We also derive an optimal estimate for the remainder term in the $C^{1,\alpha}$-case. Furthermore, if the boundary is merely Lipschitz, we show that the Hadamard formula is not valid.