The mean first passage time (MFPT) is calculated for a Brownian particle in a bounded two-dimensional domain that contains N small nonoverlapping absorbing windows on its boundary. The reciprocal of the MFPT of this narrow escape problem has wide applications in cellular biology, where it may be used as an effective first-order rate constant to describe, for example, the nuclear export of messenger RNA molecules through nuclear pores. In the asymptotic limit where the absorbing patches have

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... l measure, the method of matched asymptotic expansions is used to calculate the MFPT in an arbitrary two-dimensional domain with a smooth boundary. The theory is extended to treat the case where the boundary of the domain is piecewise smooth. The asymptotic results for the MFPT depend on the surface Neumann Green's function of the corresponding domain and its associated regular part. The known analytical formulae for the surface Neumann Green's function for the unit disk and the unit square provide explicit asymptotic approximations to the MFPT for these special domains. For an arbitrary two-dimensional domain with a smooth boundary, the asymptotic MFPT is evaluated by developing a novel boundary integral method to numerically calculate the required surface Neumann Green's function.