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

Let Ω be an open connected subset of R^n for which the Poincare' inequality holds. We consider the Dirichlet eigenvalue problem for the Laplace operator in the open subset ϕ(Ω) of R^n, where ϕ is a locally Lipschitz continuous homeomorphism of Ω onto ϕ(Ω). Then we show Lipschitz type inequalities for the reciprocals of the eigenvalues of the Rayleigh quotient ∫_ϕ(Ω)|Dv|^2dy/∫_ϕ(Ω)|v|^2dy upon variation of ϕ, which in particular yield inequalities for the proper eigenvalues of the Dirichlet Laplacian when we further assume that the imbedding of the Sobolev space W^{1,2}_0(Ω) into the space L^2(Ω) is compact. In this case, we prove the same type of inequalities for the projections onto the eigenspaces upon variation of ϕ.