{"paper":{"title":"Existence and regularity of minimizers for some spectral functionals with perimeter constraint","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Bozhidar Velichkov, Guido De Philippis","submitted_at":"2013-03-05T09:58:59Z","abstract_excerpt":"In this paper we prove that the shape optimization problem $$\\min\\left\\{\\lambda_k(\\Omega):\\ \\Omega\\subset\\R^d,\\ \\Omega\\ \\hbox{open},\\ P(\\Omega)=1,\\ |\\Omega|<+\\infty\\right\\},$$ has a solution for any $k\\in\\N$ and dimension $d$. Moreover, every solution is a bounded connected open set with boundary which is $C^{1,\\alpha}$ outside a closed set of Hausdorff dimension $d-8$. Our results are more general and apply to spectral functionals of the form $f(\\lambda_{k_1}(\\Omega),\\dots,\\lambda_{k_p}(\\Omega))$, for increasing functions $f$ satisfying some suitable bi-Lipschitz type condition."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.0968","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"}