  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
$C^{1,\alpha}$.
