Stability for the Dirichlet problem under continuous Steiner symmetrization

In this paper we prove the existence of a deformation transforming an arbit rary open set into the ball, which has the following properties: it keeps c onstant the measure, the kth eigenvalue of Laplace-Dirichlet operator is co ntinuous from the left and the first eigenvalue is decreasing. The deformat ion is given by a sequence of continuous Steiner symmetrizations, and the b ehavior of the eigenvalues is related to the stability of the Dirichlet pro blem.