{"paper":{"title":"On the minimization of Dirichlet eigenvalues","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.SP","authors_text":"M. van den Berg","submitted_at":"2014-05-01T10:35:23Z","abstract_excerpt":"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 $$J_k(c)=\\inf\\{\\lambda_k(\\Omega): \\Omega\\ \\textup{quasi-open in}\\ \\mathbb{R}^m, |\\Omega|\\le 1, \\mathcal {P}(\\Omega)\\le c \\},$$ where $c>0$, $\\lambda_k(\\Omega)$ is the $k$'th eigenvalue of the Dirichlet Laplacian acting in $L^2(\\Omega)$, $|\\Omega|$ denotes the Lebesgue measure of $\\Omega$, $\\mathcal{P}(\\Omega)$ denotes the perimeter of $\\Omega$, and where $\\mathcal{T}$ is in a suitable class set functions. The latter include for"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.0127","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"}