Optimal localization of eigenfunctions in an inhomogeneous medium

The problem of creating eigenfunctions which are localized arises in the study of photonic bandgap structures. A model problem, that of finding material inhomogeneity in a domain so that one of its Dirichlet eigenfunctions is localized, is considered in this work. The most difficult aspect, that of formulating the problem, is described, and well-posed variational problems are given. A computational approach, based on gradient descent with projection and trajectory continuation, is devised to solve the optimization problem. Numerical examples are provided which demonstrate the capability of the computational method.