Layer potential techniques for the narrow escape problem. (English) Zbl 1252.35116

Summary: The narrow escape problem consists in deriving the asymptotic expansion of the solution of a drift-diffusion equation with Dirichlet boundary condition on a small absorbing part of the boundary and 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, 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.


35C20 Asymptotic expansions of solutions to PDEs
35B40 Asymptotic behavior of solutions to PDEs
92B05 General biology and biomathematics
35P15 Estimates of eigenvalues in context of PDEs
35J61 Semilinear elliptic equations
35B25 Singular perturbations in context of PDEs
35J25 Boundary value problems for second-order elliptic equations
Full Text: DOI arXiv


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