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

@article{Larson2019AsymptoticSO,
title={Asymptotic shape optimization for Riesz means of the Dirichlet Laplacian over convex domains},
author={Simon Larson},
journal={Journal of Spectral Theory},
year={2019}
}
• Simon Larson
• Published 17 November 2016
• Mathematics
• Journal of Spectral Theory
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… 13 Citations Improved Bounds for Hermite–Hadamard Inequalities in Higher Dimensions Let $$\Omega \subset {\mathbb {R}}^n$$ Ω ⊂ R n be a convex domain and let $$f:\Omega \rightarrow {\mathbb {R}}$$ f : Ω → R be a positive, subharmonic function (i.e., $$\Delta f \ge 0$$ Δ f ≥ 0 ). 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The optimal domain is shown to be asymptotically The Lieb-Thirring inequalities: Recent results and open problems This review celebrates the generous gift by Ronald and Maxine Linde for the remodeling of the Caltech mathematics department and the author is very grateful to the editors of this volume for the ## References SHOWING 1-10 OF 59 REFERENCES On the minimization of Dirichlet eigenvalues Results are obtained for two minimization problems: $$I_k(c)=\inf \{\lambda_k(\Omega): \Omega\ \textup{open, convex in}\ \mathbb{R}^m,\ \mathcal{T}(\Omega)= c \},$$ and Lipschitz Regularity of the Eigenfunctions on Optimal Domains • Mathematics • 2015 AbstractWe study the optimal sets $${\Omega^\ast\subseteq\mathbb{R}^d}$$Ω*⊆Rd for spectral functionals of the form $${F\big(\lambda_1(\Omega),\ldots,\lambda_p(\Omega)\big)}$$F(λ1(Ω),…,λp(Ω)), which Regularity of the optimal sets for some spectral functionals • Mathematics • 2016 AbstractIn this paper we study the regularity of the optimal sets for the shape optimization problem $$\min\Big\{\lambda_{1}(\Omega)+\dots+\lambda_{k}(\Omega) : \Omega \subset {\mathbb {R}}^{d} {\rm Existence and Regularity of Minimizers for Some Spectral Functionals with Perimeter Constraint • Physics, Mathematics • 2013 In this paper we prove that the shape optimization problem$$\min \bigl\{\lambda_k(\varOmega):\ \varOmega\subset \mathbb{R}^d,\ \varOmega\ \hbox{open},\ P(\varOmega)=1,\ |\varOmega|<+\infty \bigr\}, Asymptotic Behaviour of Cuboids Optimising Laplacian Eigenvalues • Mathematics • 2017 We prove that in dimension $$n \ge 2$$n≥2, within the collection of unit-measure cuboids in $$\mathbb {R}^n$$Rn (i.e. domains of the form $$\prod _{i=1}^{n}(0, a_n)$$∏i=1n(0,an)), any sequence of Maximizing Riesz means of anisotropic harmonic oscillators We consider problems related to the asymptotic minimization of eigenvalues of anisotropic harmonic oscillators in the plane. 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Recent Existence Results for Spectral Problems
In this survey we present the new techniques developed for proving existence of optimal sets when one minimizes functionals depending on the eigenvalues of the Dirichlet Laplacian with a measure