The laplacian in regions with many small obstacles: Fluctuations around the limit operator

We consider the Laplacianδm in ℝ3 (or in a bounded region of ℝ3) with Dirichlet boundary conditions on the surfaces of some identical (small) neighborhoods ofm randomly distributed points, in the limit whenm goes to infinity and their linear size decreases as 1/m. We give here a stronger form of the result showing the convergence of the above operator toδ − C(x), whereC(x) is the limit density of electrostatic capacity of the “obstacles.” In particular results on the rate of convergence and on the fluctuations ofδm around the limit operator are given.