A global Lipschitz continuity result for a domain-dependent Neumann eigenvalue problem for the Laplace operator

Let ­ be an open connected subset of R^n of finite measure for which the Poincare'-Wirtinger inequality holds. We consider the Neumann eigenvalue problem for the Laplace operator in the open subset \phi(\Omega) of R^n, where \phi(\Omega) is a locally Lipschitz continuous homeomorphism of \Omega ­ onto \phi(\Omega). Then we show Lipschitz type inequalities for the reciprocals of the eigenvalues delivered by the Rayleigh quotient. Then we further assume that the imbedding of the Sobolev space W^{1,2}(­\Omega) into the space L^2(\Omega) is compact, and we prove the same type of inequalities for the projections onto the eigenspaces upon variation of \phi.