This paper considers how the eigenvalues of the Neumann problem for an
elliptic operator depend on the domain. The proximity of two domains is
measured in terms of the norm of the difference between the two resolvents
corresponding to the reference domain and the perturbed domain, and the size of
eigenfunctions outside the intersection of the two domains. This construction
enables the possibility of comparing both nonsmooth domains and domains with
different topology. An abstract framework is presented, where the main result
is an asymptotic formula where the remainder is expressed in terms of the
proximity quantity described above when this is relatively small. We consider
two applications: the Laplacian in both C1,α and Lipschitz domains.
For the C1,α case, an asymptotic result for the eigenvalues is given
together with estimates for the remainder, and we also provide an example which
demonstrates the sharpness of our obtained result. For the Lipschitz case, the
proximity of eigenvalues is estimated