Optimizing the principal eigenvalue of the Laplacian in a sphere with interior traps

Abstract

The method of matched asymptotic expansions is used to calculate a two-term asymptotic expansion for the principal eigenvalue λ ( ε ) of the Laplacian in a three-dimensional domain Ω with a reflecting boundary that contains N interior traps of asymptotically small radii. In the limit of small trap radii ε → 0 , this principal eigenvalue is inversely proportional to the average mean first passage time (MFPT), defined as the expected time required for a Brownian particle undergoing free diffusion, and with a uniformly distributed initial starting point in Ω , to be captured by one of the localized traps. The coefficient of the second-order term in the asymptotic expansion of λ ( ε ) is found to depend on the spatial locations of the traps inside the domain, together with the Neumann Green’s function for the Laplacian. For a spherical domain, where this Green’s function is known analytically, the discrete variational problem of maximizing the coefficient of the second-order term in the expansion of λ ( ε ) , or correspondingly minimizing the average MFPT, is studied numerically by global optimization methods for N ≤ 20 traps. Moreover, the effect on the average MFPT of the fragmentation of the trap set is shown to be rather significant for a fixed trap volume fraction when N is not too large. Finally, the method of matched asymptotic expansions is used to calculate the splitting probability in a three-dimensional domain, defined as the probability of reaching a specific target trap from an initial source point before reaching any of the other traps. Some examples of the asymptotic theory for the calculation of splitting probabilities are given for a spherical domain.