Vol. 8, No. 7, 2015

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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
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 ${C}^{1,\alpha }$.

Keywords
Hadamard formula, domain variation, asymptotics of eigenvalues, Neumann problem, $C^1$-domains
Mathematical Subject Classification 2010
Primary: 35P05, 47A55, 47A75, 49R05
Milestones
Accepted: 3 September 2015
Published: 18 September 2015