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

The method of matched asymptotic expansions is used to calculate a two-term asymptotic expansion for the principal eigenvalue λ(ε)λ(ε) of the Laplacian in a three-dimensional domain ΩΩ with a reflecting boundary that contains NN interior traps of asymptotically small radii. In the limit of small trap radii ε→0ε→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 ΩΩ, to be captured by one of the localized traps. The coefficient of the second-order term in the asymptotic expansion of λ(ε)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)λ(ε), or correspondingly minimizing the average MFPT, is studied numerically by global optimization methods for N≤20N≤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 NN 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.