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2013 | OriginalPaper | Chapter

8. Narrow Escape in ${\mathbb{R}}^{3}$

Author : Zeev Schuss

Published in: Brownian Dynamics at Boundaries and Interfaces

Publisher: Springer New York

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Abstract

The NET problem in three dimensions is more complicated than that in two dimension, primarily because the singularity of Neumann’s function for a regular domain is more complicated than (7.1).

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previous chapter Narrow Escape in

For nonsmooth p ₀, the integral is not uniformly bounded. For example, for $p_{0} =\delta (\mbox{ $\boldsymbol{x}$} -\mbox{ $\boldsymbol{x}$}_{0})$, we have $\int _{\Omega }N(\mbox{ $\boldsymbol{x}$},\mbox{ $\boldsymbol{y}$})p_{0}(\mbox{ $\boldsymbol{y}$})\,\mathrm{d}\mbox{ $\boldsymbol{y}$} = N(\mbox{ $\boldsymbol{x}$},\mbox{ $\boldsymbol{x}$}_{0})$, which becomes singular as $\mbox{ $\boldsymbol{x}$} \rightarrow \mbox{ $\boldsymbol{x}$}_{0}$. However, this is an integrable singularity, and as such it does not affect the leading-order asymptotics in δ.

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Title: Narrow Escape in
Author: Zeev Schuss
Publisher: Springer New York
Book: Brownian Dynamics at Boundaries and Interfaces
Print ISBN: 978-1-4614-7686-3

Electronic ISBN: 978-1-4614-7687-0

Copyright Year: 2013
DOI: https://doi.org/10.1007/978-1-4614-7687-0_8

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