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

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 any $\Lambda \ge 0$ and $\gamma \ge 0$ let ${\Omega }_{\Lambda ,\gamma }(\mathcal{A})$ denote any extremal set of the shape optimization problem

where $\mathcal{A}$ is an admissible family of convex domains in ${\mathbb{R}}^n$. If $\gamma \ge 1$ and $\{{\Lambda }_j{\}}_{j\ge 1}$ is a positive sequence tending to infinity we prove that $\{{\Omega }_{{\Lambda }_j,\gamma }(\mathcal{A}){\}}_{j\ge 1}$ 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.

A correction to this paper is available.