Asymptotics of Hadamard Type for Eigenvalues of the Neumann Problem on $C^1$-domains for Elliptic Operators

TL;DR

This paper derives asymptotic formulas for the eigenvalues of the Neumann problem for elliptic operators on $C^1$-domains, extending previous Lipschitz domain results and analyzing $C^1$-perturbations.

Contribution

It provides the first asymptotic eigenvalue formulas for Neumann problems on $C^1$-domains, expanding the understanding beyond Lipschitz cases.

Findings

Derived asymptotic formulas for eigenvalues on $C^1$-domains.

Extended previous Lipschitz domain results to $C^1$-domains.

Analyzed eigenvalue behavior under $C^1$-perturbations.

Abstract

This article investigates how the eigenvalues of the Neumann problem for an elliptic operator depend on the domain in the case when the domains involved are of class $C^1$. We consider the Laplacian and use results developed previously for the corresponding Lipschitz case. In contrast with the Lipschitz case however, in the $C^1$-case we derive an asymptotic formula for the eigenvalues when the domains are of class $C^1$. Moreover, as an application we consider the case of a $C^1$-perturbation when the reference domain is of class $C^{1,\alpha}$.