Asymptotics of Hadamard Type for Eigenvalues of the Neumann Problem on C¹-domains for Elliptic Operators

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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.