Global minimizing domains for the first eigenvalue of an elliptic operator with non-constant coefficients

Abstract

We consider an elliptic operator, in divergence form, that is a uniformly elliptic matrix. We describe the behavior of every sequence of domains which minimizes the first Dirichlet eigenvalue over a family of fixed measure domains of ℝN. The existence of minimizers is proved in some particular situations, for example when the operator is periodic.