The effect of a singular perturbation on nonconvex variational problems

Abstract

We study the effect of a singular perturbation on certain nonconvex variational problems. The goal is to characterize the limit of minimizers as some perturbation parameter ɛ → 0. The technique utilizes the notion of “Γ-convergence” of variational problems developed by De Giorgi. The essential idea is to identify the first nontrivial term in an asymptotic expansion for the energy of the perturbed problem. In so doing, one characterizes the limit of minimizers as the solution of a new variational problem. For the cases considered here, these new problems have a simple geometric nature involving minimal surfaces and geodesics.