Asymptotics of Hadamard type for eigenvalues of the Neumann problem on C1-domains for elliptic operators

Thim, Johan

doi:10.2140/apde.2015.8.1695

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Asymptotics of Hadamard type for eigenvalues of the Neumann problem on $C^1$-domains for elliptic operators

Johan Thim

Vol. 8 (2015), No. 7, 1695–1706

DOI: 10.2140/apde.2015.8.1695

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 $C^{1, α}$ .

Keywords

Hadamard formula, domain variation, asymptotics of eigenvalues, Neumann problem, $C^1$-domains

Mathematical Subject Classification 2010

Primary: 35P05, 47A55, 47A75, 49R05

Milestones

Received: 18 December 2014

Accepted: 3 September 2015

Published: 18 September 2015

Authors

Johan Thim
Department of Mathematics Linköping University SE-58183 Linköping Sweden

Vol. 8, No. 7, 2015