# Convex combinations of low eigenvalues, Fraenkel asymmetries and attainable sets

created by mazzoleni on 03 Dec 2015
modified on 28 Jul 2016

[BibTeX]

Accepted Paper: ESAIM: COCV

Inserted: 3 dec 2015
Last Updated: 28 jul 2016

Year: 2015

Abstract:

We consider the problem of minimizing convex combinations of the first two eigenvalues of the Dirichlet-Laplacian among open set of $R^N$ of fixed measure. We show that, by purely elementary arguments, based on the minimality condition, it is possible to obtain informations on the geometry of the minimizers of convex combinations: we study, in particular, when these minimizers are no longer convex, and the optimality of balls. As an application of our results we study the boundary of the attainable set for the Dirichlet spectrum.

Our techniques involve symmetry results similar to those by Serrin, explicit constants in quantitative inequalities, as well as a purely geometrical problem: the minimization of the Fraenkel 2-asymmetry among convex sets of fixed measure.