Lipschitz type inequalities for a domain dependent Neumann eigenvalue problem for the Laplace operator

In this paper we announce some results concerning the dependence of Neumann eigenvalues and eigenvectors of the Laplace operator upon domain perturbation. Let \Omega be an open connected subset of R^N of finite measure for which the Sobolev space W^{1,2}(\Omega) is compactly embedded into the space L^2(\Omega). We consider the Laplace operator with Neumann boundary conditions in a class of domains \phi(\Omega), where \phi is a locally Lipschitz continuous homeomorphism of \Omega onto the subset \phi(\Omega) of R^N. Then we present Lipschitz type inequalities for the reciprocals of the eigenvalues and for the projections onto the eigenspaces upon variation of \phi.