On the minimization of Dirichlet eigenvalues of the Laplace operator

We study variational problems of the form inf{lk(W): W open in \mathbbRm, T(W) £ 1 },infk(): open in Rm T()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 minimizer for k=1. Upper bounds are obtained for the number of components of any bounded minimizer if T satisfies a scaling relation. For example, we show that if T is Lebesgue measure and if k≤m+1 then any bounded minimizer 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.