On an Elaboration of M. Kac's theorem concerning eigenvalues of $- \Delta $ in a region with randomly distributed small obstacles

Abstract

We removem-balls of centersw1,...,wm with the same radius α/m from a bounded domain Ω inR3 with smooth boundary γ. Let μk(α/m;w(m)) denote thek-th eigenvalue of the Laplacian in Ω/m-balls under the Dirichlet condition. We consider μk(α/m;w(m)) as a random variable on a probability space (w1,...,wm)∈Ω × ... × Ω and we examine a precise behaviour of μk(α/m;w(m)) asm → ∞. We give an elaboration of. M. Kac's theorem.