Layer potential techniques for the narrow escape problem

Abstract

The narrow escape problem consists in deriving the asymptotic expansion of the solution of a drift-diffusion equation with the Dirichlet boundary condition on a small absorbing part of the boundary and the Neumann boundary condition on the remaining reflecting boundaries. Using layer potential techniques, we rigorously find high-order asymptotic expansions of such solutions. The asymptotic formula explicitly exhibits the nonlinear interaction of many small absorbing targets. Based on the asymptotic theory for eigenvalue problems developed in Ammari et al. (2009) [3], we also construct high-order asymptotic formulas for the perturbation of eigenvalues of the Laplace and the drifted Laplace operators for mixed boundary conditions on large and small pieces of the boundary.