Some estimates for the higher eigenvalues of sets close to the ball

In this paper we investigate the behavior of the eigenvalues of the Dirichlet Laplacian on sets in RN whose first eigenvalue is close to the one of the ball with the same volume. In particular in our main Theorem 1.1 we prove that, for all k 2 N, there is a positive constant C D C.k; N / such that for every open set Ω ⊆ RN with unit measure and with λ1.Ω/ not excessively large one has jλk.Ω/ - λk.B/j ≤ C.λ1.Ω/ - λ1.B//β; λk.B/ - λk.Ω/ ≤ Cd.Ω/β'; where d.Ω/ is the Fraenkel asymmetry of Ω, and where β and β' are explicit exponents, not depending on k nor on N; for the special case N D 2, a better estimate holds.