Diffusion escape through a cluster of small absorbing windows

We study the first eigenvalue of the Laplace equation in a bounded domain in {{\bb R}}^d (d=2,3) with mixed Neumann-Dirichlet (Zaremba) boundary conditions. The Neumann condition is imposed on most of the boundary and the Dirichlet boundary consists of a cluster of small windows. When the windows are well separated the first eigenvalue is asymptotically the sum of eigenvalues of mixed problems with a single Dirichlet window. However, when two or more Dirichlet windows cluster tightly together they interact nonlinearly. We compare our asymptotic approximation of the eigenvalue to the escape rate of simulated Brownian particles through the small windows.