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

TL;DR

This paper investigates the asymptotic behavior of extremal convex domains that optimize Riesz means of the Dirichlet Laplacian spectrum, showing convergence to perimeter-minimizing shapes like regular polygons.

Contribution

It characterizes the limits of shape optimization problems for Riesz means of the Dirichlet Laplacian in convex domains as perimeter minimizers, with explicit results for polygons.

Findings

Sequences of extremal domains are bounded and have convergent subsequences.

Limit shapes are characterized as perimeter minimizers within the admissible family.

For polygons with a fixed number of faces, extremal domains converge to regular polygons.

Abstract

For $\Omega \subset \mathbb{R}^n$, a convex and bounded domain, we study the spectrum of $-\Delta_\Omega$ the Dirichlet Laplacian on $\Omega$. For $\Lambda\geq0$ and $\gamma \geq 0$ let $\Omega_{\Lambda, \gamma}(\mathcal{A})$ denote any extremal set of the shape optimization problem $$ \sup\{ \mathrm{Tr}(-\Delta_\Omega-\Lambda)_-^\gamma: \Omega \in \mathcal{A}, |\Omega|=1\}, $$ where $\mathcal{A}$ is an admissible family of convex domains in $\mathbb{R}^n$. If $\gamma \geq 1$ and $\{\Lambda_j\}_{j\geq1}$ is a positive sequence tending to infinity we prove that $\{\Omega_{\Lambda_j, \gamma}(\mathcal{A})\}_{j\geq1}$ is a bounded sequence, and hence contains a convergent subsequence. Under an additional assumption on $\mathcal{A}$ we characterize the possible limits of such subsequences as minimizers of the perimeter among domains in $\mathcal{A}$ of unit measure. For instance if $\mathcal{A}$ is the set of all convex polygons with no more than $m$ faces, then $\Omega_{\Lambda, \gamma}$ converges, up to rotation and translation, to the regular $m$-gon. This is a revised version of the paper published in the Journal of Spectral Theory (2019) which has been updated in accordance with an erratum published in 2021. The results of the paper remain unchanged, but the proofs of Theorem 2.4 and Corollary 5.3 have been amended.