{"paper":{"title":"Asymptotic shape optimization for Riesz means of the Dirichlet Laplacian over convex domains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.OC"],"primary_cat":"math.SP","authors_text":"Simon Larson","submitted_at":"2016-11-17T13:55:06Z","abstract_excerpt":"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 $$\n  \\sup\\{ \\mathrm{Tr}(-\\Delta_\\Omega-\\Lambda)_-^\\gamma: \\Omega \\in \\mathcal{A}, |\\Omega|=1\\}, $$ 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}(\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.05680","kind":"arxiv","version":5},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1611.05680/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}