{"paper":{"title":"Regularity of the optimal sets for some spectral functionals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","math.OC"],"primary_cat":"math.AP","authors_text":"Bozhidar Velichkov, Dario Mazzoleni, Susanna Terracini","submitted_at":"2016-09-05T17:35:51Z","abstract_excerpt":"In 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,\\ \\text{open}\\ ,\\ |\\Omega|=1\\Big\\}, \\] where $\\lambda_1(\\cdot),\\dots,\\lambda_k(\\cdot)$ denote the eigenvalues of the Dirichlet Laplacian and $|\\cdot|$ the $d$-dimensional Lebesgue measure. We prove that the topological boundary of a minimizer $\\Omega_k^*$ is composed of a relatively open regular part which is locally a graph of a $C^{1,\\alpha}$ function and a closed singular part, which is empty if $d<d^*$, contains at"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.01231","kind":"arxiv","version":3},"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"}