Optimal spatially-dependent diffusion tensors

Abstract

We consider the spatially-dependent diffusion tensors that maximize the rate of convergence to equilibrium of solutions to the heat equation in inhomogeneous media. We formulate a variational problem for the optimal spectral gap, subject to an Lp constraint on the diffusion tensors, and solve a relaxed version of this variational problem. This provides an upper bound for the optimal spectral gap. The solution is characterized in terms of the extremals of Sobolev and Poincaré type inequalities. To illustrate this result, we give a presentation of the one-dimensional case. In this case, we can adapt the known closed solution of a related eigenvalue problem to write the solution to the relaxed variational problem, arriving at an expression for the upper bound on the optimal spectral gap. The sharpness of this upper bound is then investigated by comparison to numerical simulations.