A real analyticity result for symmetric functions of the eigenvalues of a domain dependent Dirichlet problem for the Laplace operator

Let ­ \Omega be an open connected subset of R^n for which the imbedding of the Sobolev space W^{1,2}_0(\Omega) into the space L^2(­\Omega) is compact. We consider the Dirichlet eigenvalue problem for the Laplace operator in the open subset \phi(\Omega) of R^n, where \phi is a Lipschitz continuous homeomorphism of ­ onto \phi(\Omega). Then we prove a result of real analytic dependence for symmetric functions of the eigenvalues upon variation of \phi.