Layer Potential Techniques for the Narrow Escape Problem

2010/03/11 by Habib Ammari, Ammari, Habib, Hyeonbae Kang +4 · 2 citations
Biochemistry, Genetics and Molecular Biology · Computer Science · Physics and Astronomy · #35B40 #92B05 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Diffusion and Search Dynamics #FOS: Mathematics #stochastic dynamics and bifurcation

paper · pdf · doi:10.48550/arxiv.1003.2275

openalex publication_date 2010/03/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 \citebook, 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.

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