An asymptotic shape optimization problem for Riesz means of Laplacian eigenvalues

TL;DR

This paper investigates the asymptotic behavior of Riesz means of Laplacian eigenvalues in convex sets, showing convergence to a ball and exploring optimization over unions of convex sets.

Contribution

It provides new insights into the asymptotic shape optimization of Riesz means and extends results to unions of convex sets.

Findings

Optimizing sets converge to a ball as the cut-off parameter increases

Results include optimization over disjoint unions of convex sets

The study focuses on a specific range of Riesz exponents

Abstract

We review our recent results on the problem of optimizing Riesz means of Laplace eigenvalues among convex sets of given measure in the regime where the cut-off parameter in the definition of the Riesz means tends to infinity. We show that for a certain range of Riesz exponents, the optimizing sets converge to a ball. We also present some new results where we optimize over disjoint unions of convex sets.