{"paper":{"title":"Regularity of Minimizers of Shape Optimization Problems involving Perimeter","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Bozhidar Velichkov (CVGI), Guido De Philippis (UMPA-ENSL), IRMAR), Jimmy Lamboley (CEREMADE), Michel Pierre (ENS Rennes","submitted_at":"2016-05-20T11:29:18Z","abstract_excerpt":"We prove existence and regularity of optimal shapes for the problem$$\\min\\Big\\{P(\\Omega)+\\mathcal{G}(\\Omega):\\ \\Omega\\subset D,\\ |\\Omega|=m\\Big\\},$$where $P$ denotes the perimeter, $|\\cdot|$ is the volume, and the functional $\\mathcal{G}$ is either one of the following:\\textless{}ul\\textgreater{}\\textless{}li\\textgreater{} the Dirichlet energy $E\\_f$, with respect to a (possibly sign-changing) function $f\\in L^p$;\\textless{}/li\\textgreater{}\\textless{}li\\textgreater{}a spectral functional of the form $F(\\lambda\\_{1},\\dots,\\lambda\\_{k})$, where $\\lambda\\_k$ is the $k$th eigenvalue of the Dirich"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.06294","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"}