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

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.Comment: Accepted and final version. 29 page