{"paper":{"title":"Asymptotic behaviour of cuboids optimising Laplacian eigenvalues","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.SP","authors_text":"Katie Gittins, Simon Larson","submitted_at":"2017-03-29T21:39:20Z","abstract_excerpt":"We prove that in dimension $n \\geq 2$, within the collection of unit measure cuboids in $\\mathbb{R}^n$ (i.e. domains of the form $\\prod_{i=1}^{n}(0, a_n)$), any sequence of minimising domains $R_k^\\mathcal{D}$ for the Dirichlet eigenvalues $\\lambda_k$ converges to the unit cube as $k \\to \\infty$. Correspondingly we also prove that any sequence of maximising domains $R_k^\\mathcal{N}$ for the Neumann eigenvalues $\\mu_k$ within the same collection of domains converges to the unit cube as $k\\to \\infty$. For $n=2$ this result was obtained by Antunes and Freitas in the case of Dirichlet eigenvalues "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.10249","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}