An optimization problem for the fundamental eigenvalue λ₀ of the Laplacian in a planar simply-connected domain that contains N small identically-shaped holes, each of a small radius ε << 1, is considered. A Neumann boundary condition is imposed on the outer boundary of the domain and a Dirichlet condition is imposed on the boundary of each of the holes. For small hole radii ε, we derive an asymptotic expansion for λ₀ in terms of certain properties of the Neumann Green's function for the Laplacian. 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 the eigenvalue with respect to the hole locations. This optimization problem gives the optimum places to insert localized traps inside a domain in order to minimize the lifetime of a Brownian particle. This eigenvalue optimization problem is also shown to be closely related to the problem of determining equilibrium vortex configurations in the Ginzburg-Landau theory of superconductivity. Finally we discuss a few interesting open problems related to eigenvalue optimization, including the problem of inserting small windows on the boundary of the domain to optimize heat loss and the problem in theoretical ecology of re-seeding a species that is on the verge of extinction.

This is joint work with Theodore Kolokolnikov (Dalhousie University) and Michèle Titcombe (University of Montreal).