Optimizing the fundamental Neumann eigenvalue for the Laplacian in a domain with small traps

An optimization problem for the fundamental eigenvalue gimel(0) of the Laplacian in a planar simply-connected domain that contains N small identically-shaped holes, each of radius epsilon << 1, is considered. The boundary condition on the domain is assumed to be of Neumann type, and a Dirichlet condition is imposed on the boundary of each of the holes. As an application, the reciprocal of the fundamental eigenvalue gimel(0) is proportional to the expected lifetime for Brownian motion in a domain with a reflecting boundary that contains N small traps. For small hole radii epsilon, a two-term asymptotic expansion for gimel(0) is derived in terms of certain properties of the Neumann Green's function for the Laplacian. Only the second term in this expansion depends on the locations x(i), for i = 1, ..., N, of the small holes. For the unit disk, ring-type configurations of holes are constructed to optimize this term with respect to the hole locations. The results yield hole configurations that asymptotically optimize gimel(0). For a class of symmetric dumbbell-shaped domains containing exactly one hole, it is shown that there is a unique hole location that maximizes gimel(0). For an asymmetric dumbbell-shaped domain, it is shown that there can be two hole locations that locally maximize gimel(0). This optimization problem is found to be directly related to an oxygen transport problem in skeletal muscle tissue, and to determining equilibrium locations of spikes to the Gierer-Meinhardt reaction-diffusion model. It is also closely related to the problem of determining equilibrium vortex configurations within the context of the Ginzburg-Landau theory of superconductivity.