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

An optimization problem for the fundamental eigenvalue (lambda)0 of the Laplacian in a planar simplyconnected 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 (lambda)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 (lambda)0 is derived in terms of certain properties of the Neumann Greenʹs 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 (lambda)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 (lambda)0. For an asymmetric dumbbell-shaped domain, it is shown that there can be two hole locations that locally maximize (lambda)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.