 
Summary: Layer Potential Techniques for the Narrow Escape Problem
Habib Ammari
Kostis Kalimeris
Hyeonbae Kang
Hyundae Lee
Abstract
The narrow escape problem consists of deriving the asymptotic expansion of the
solution of a driftdiffusion equation with the Dirichlet boundary condition on a small
absorbing part of the boundary and the Neumann boundary condition on the re
maining reflecting boundaries. Using layer potential techniques, we rigorously find
highorder asymptotic expansions of such solutions. The asymptotic formula explic
itly exhibit the nonlinear interaction of many small absorbing targets. Based on the
asymptotic theory for eigenvalue problems developed in [2], 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.
Mathematics Subject Classification (MSC2000): 35B40, 92B05
Keywords: narrow escape problem, mean sojourn time, driftdiffusion, asymptotic expansion, small hole,
mixed boundary value problem, clustered targets
1 Introduction
