Layer Potential Techniques for the Narrow Escape Problem

TL;DR

This paper develops high-order asymptotic expansions for solutions and eigenvalues of drift-diffusion equations with small absorbing regions, using layer potential techniques to analyze the narrow escape problem.

Contribution

It introduces rigorous high-order asymptotic formulas for solutions and eigenvalues in the narrow escape problem, highlighting nonlinear interactions of small targets.

Findings

Explicit high-order asymptotic expansions for solutions.

High-order formulas for eigenvalues of Laplace and drifted Laplace operators.

Analysis of nonlinear interactions among small absorbing targets.

Abstract

The narrow escape problem consists of 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. We explicitly show the nonlinear interaction of many small absorbing targets. Based on the asymptotic theory for eigenvalue problems developed in \cite{book}, we also construct high-order asymptotic formulas for eigenvalues of the Laplace and the drifted Laplace operators for mixed boundary conditions on large and small pieces of the boundary.