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Abstract. The mean first passage time (MFPT) is calculated for a Brownian particle in a spherical domain in R 3 that contains N small nonoverlapping absorbing windows, or traps, on its boundary. For the unit sphere, the method of matched asymptotic expansions is used to derive an explicit three-term asymptotic expansion for the MFPT for the case of N small locally circular absorbing windows. The third term in this expansion, not previously calculated, depends explicitly on the spatial configuration of the absorbing windows on the boundary of the sphere. The threeterm asymptotic expansion for the average MFPT is shown to be in very close agreement with full numerical results. The average MFPT is shown to be minimized for trap configurations that minimize a certain discrete variational problem. This variational problem is closely related to the well-known optimization problem of determining the minimum energy configuration for N repelling point charges on the unit sphere. Numerical results, based on global optimization methods, are given for both the optimum discrete energy and the arrangements of the centers {x 1 , . . . , x N } of N circular traps on the boundary of the sphere. A scaling law for the optimum discrete energy, valid for N 1, is also derived.Key words. narrow escape, mean first passage time, matched asymptotic expansions, surface Neumann Green's functions, discrete variational problem, logarithmic switchback terms AMS subject classifications. 35B25, 35C20, 35P15, 35J05, 35J08DOI. 10.1137/100782620 1. Introduction. The narrow escape problem concerns the motion of a Brownian particle confined in a bounded domain Ω ∈ R d (d = 2, 3) whose boundary ∂Ω = ∂Ω r ∪ ∂Ω a is almost entirely reflecting (∂Ω r ), except for small absorbing windows, or traps, labeled collectively by ∂Ω a , through which the particle can escape. Denoting the trajectory of the Brownian particle by X(t), the mean first passage time (MFPT) v(x) is defined as the expectation value of the time τ taken for the Brownian particle to become absorbed somewhere in ∂Ω a starting initially fromThe calculation of v(x) becomes a narrow escape problem in the limit when the measure of the absorbing set |∂Ω a | = O(ε d−1 ) is asymptotically small, where 0 < ε 1 measures the dimensionless radius of an absorbing window. Since the MFPT diverges as ε → 0, the calculation of the MFPT v(x) constitutes a singular perturbation problem.The narrow escape problem has many applications in biophysical modeling (see [2], [16], [19], [39] and the references therein). For the case of a two-dimensional domain, the narrow escape problem has been studied with a variety of analytical methods in [19], [42], [43], [20], and Part I of this paper [30]. In this paper, we use

This paper considers the effect of three types of perturbations of large magnitude but small extent on a class of linear eigenvalue problems for elliptic partial differential equations in bounded or unbounded domains. The perturbations are the addition of a function of small support and large magnitude to the differential operator, the removal of a small subdomain from the domain of a problem with the imposition of a boundary condition on the boundary of the resulting hole, and a large alteration of the boundary condition on a small region of the boundary of the domain. For each of these perturbations, the eigenvalues and eigenfunctions for the perturbed problem are constructed by the method of matched asymptotic expansions for small, where is a measure of the extent of the perturbation. In some special cases, the asymptotic results are shown to agree well with exact results. The asymptotic theory is then applied to determine the exit time distribution for a particle undergoing Brown|an motion inside a container having reflecting walls perforated by many small holes.

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The dynamical behavior of multi-spot solutions in a two-dimensional domain Ω is analyzed for the two-component Schnakenburg reaction-diffusion model in the singularly perturbed limit of small diffusivity ε for one of the two components. In the limit ε → 0, a quasi-equilibrium spot pattern in the region away from the spots is constructed by representing each localized spot as a logarithmic singularity of unknown strength Sj for j = 1, . . . , K at unknown spot locations xj ∈ Ω for j = 1, . . . , K. A formal asymptotic analysis, which has the effect of summing infinite logarithmic series in powers of −1/ log ε, is then used to derive an ODE differential algebraic system (DAE) for the collective coordinates Sj and xj for j = 1, . . . , K, which characterizes the slow dynamics of a spot pattern. This DAE system involves the Neumann Green's function for the Laplacian. By numerically examining the stability thresholds for a single spot solution, a specific criterion in terms of the source strengths Sj , for j = 1, . . . , K, is then formulated to theoretically predict the initiation of a spot-splitting event. The analytical theory is illustrated for spot patterns in the unit disk and the unit square, and is compared with full numerical results computed directly from the Schnakenburg model.