# Asymptotic shape optimization for Riesz means of the Dirichlet Laplacian over convex domains

### Simon Larson

KTH Royal Institute of Technology, Stockholm, Sweden

## Abstract

For $Ω⊂R_{n}$, a convex and bounded domain, we study the spectrum of $−Δ_{Ω}$ the Dirichlet Laplacian on $Ω$. For any $Λ≥0$ and $γ≥0$ let $Ω_{Λ,γ}(A)$ denote any extremal set of the shape optimization problem

where $A$ is an admissible family of convex domains in $R_{n}$. If $γ≥1$ and ${Λ_{j}}_{j≥1}$ is a positive sequence tending to infinity we prove that ${Ω_{Λ_{j},γ}(A)}_{j≥1}$ is a bounded sequence, and hence contains a convergent subsequence. Under an additional assumption on $A$ we characterize the possible limits of such subsequences as minimizers of the perimeter among domains in $A$ of unit measure. For instance if $A$ is the set of all convex polygons with no more than $m$ faces, then $Ω_{Λ,γ}$ converges, up to rotation and translation, to the regular $m$-gon.

## Cite this article

Simon Larson, Asymptotic shape optimization for Riesz means of the Dirichlet Laplacian over convex domains. J. Spectr. Theory 9 (2019), no. 3, pp. 857–895

DOI 10.4171/JST/265