Optimization of nonlinear eigenvalues under measure or perimeter constraint

Abstract

In this paper we recall some recent results about variational eigenvalues of the p-Laplacian, we show new applications and point out some open problems. We focus on the continuity properties of the eigenvalues under the gamma_p-convergence of capacitary measures, which are needed to prove existence results for the minimization of nonlinear eigenvalues in the class of p-quasi open sets contained in a box under a measure constraint. Finally, the new contribution of this paper is to show that these continuity results can be employed to prove existence of minimizers for nonlinear eigenvalues among measurable sets contained in a box and under a perimeter constraint.