  The dependence on the domain is studied for the Dirichlet eigenvalues of an
elliptic operator considered in bounded domains. Their proximity is measured by
a norm of the difference of two orthogonal projectors corresponding to the
reference domain and the perturbed one; this allows to compare domains that
have non-smooth boundaries and different topology. The main result is an
asymptotic formula in which the remainder is evaluated in terms of this
quantity. As an application, the stability of eigenvalues is estimated by
virtue of integrals of squares of the gradients of eigenfunctions for elliptic
problems in different domains. It occurs that these stability estimates imply
well-known inequalities for perturbed eigenvalues.
