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

arXiv: https://arxiv.org/abs/1512.00993 · HTML: https://mlsanity.com/paper/1512.00993/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Dec 2015.

## Built on by (most cited first, 1 of 1)

- [Numerical exploration of the range of shape functionals using neural networks](https://mlsanity.com/paper/2602.14881/) (2026)

## Abstract (arXiv)

We consider the problem of minimizing convex combinations of the first two eigenvalues of the Dirichlet-Laplacian among open sets of RN 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 `a la 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.