Vol. 8, No. 7, 2015

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
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 C1. We consider the Laplacian and use results developed previously for the corresponding Lipschitz case. In contrast with the Lipschitz case, however, in the C1-case we derive an asymptotic formula for the eigenvalues when the domains are of class C1. Moreover, as an application we consider the case of a C1-perturbation when the reference domain is of class C1,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