We consider the variational problemis the kth eigenvalue of the Dirichlet Laplacian acting in L 2 (Ω) and |Ω| is the Lebesgue measure of Ω. We investigate for which values of α, β every minimiser is connected.

We study variational problems of the form inf{λ k (Ω) : Ω open in R m , T (Ω) ≤ 1}, where λ k (Ω) is the k'th eigenvalue of the Dirichlet Laplacian acting in L 2 (Ω), and where T is a non-negative set function defined on the open sets in R m , which is invariant under isometries, additive on disjoint families of open sets, and is such that the ball with T (B) = 1 is a minimiser for k = 1. Upper bounds are obtained for the number of components of any bounded minimiser if T satisfies a scaling relation. For example we show that if T is Lebesgue measure and if k ≤ m + 1 then any bounded minimiser has at most 7 components. We also consider variational problems over open sets Ω in R m involving the (m − 1) -dimensional Hausdorff measure of ∂Ω. Mathematics Subject Classification (2000): 49Q10; 49R50; 35P15.

The goal of this paper is to study the Dirichlet eigenvalues of bounded domains Ω ⊂ Ω ′ . With a local spectral stability requirement on Ω, we show that the difference of the Dirichlet eigenvalues of Ω ′ and Ω is explicitly controlled from above in terms of the first eigenvalue of Ω ′ \ Ω and of geometric constants depending on the inner domain Ω. In particular, Ω ′ can be an arbitrary bounded domain.