Questions of flux regulation in biological cells raise renewed interest inthe narrow escape problem. The often inadequate expansions of the narrow escapetime are due to a not so well known fact that the boundary singularity ofGreen's function for Poisson's equation with Neumann and mixedDirichlet-Neumann boundary conditions in three-dimensions contains alogarithmic singularity. Using this fact, we find the second term in theexpansion of the narrow escape time and in the expansion of the principaleigenvalue of the Laplace equation with mixed Dirichlet-Neumann boundaryconditions, with small Dirichlet and large Neumann parts. We also find theleakage flux of Brownian particles that diffuse from a source to an absorbingtarget on a reflecting boundary of a domain, if a small perforation is made inthe reflecting boundary.
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