{"paper":{"title":"Multiphase shape optimization problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Bozhidar Velichkov, Dorin Bucur","submitted_at":"2013-10-09T12:12:17Z","abstract_excerpt":"This paper is devoted to the analysis of multiphase shape optimization problems, which can formally be written as $\\min\\Big\\{{g}(F_1(\\Omega_1),\\dots,F_h(\\Omega_h))+ m\\vert\\,\\bigcup_{i=1}^h\\Omega_i\\vert :\\ \\Omega_i\\subset D,\\ \\Omega_i\\cap \\Omega_j =\\emptyset\\Big\\},$ where $D\\subset\\mathcal{R}^d$ is a given bounded open set, $\\vert\\Omega_i\\vert$ is the Lebesgue measure of $\\Omega_i$ and $m$ is a positive constant. For a large class of such functionals, we analyse qualitative properties of the cells $\\Omega_i$ and the interaction between them. Each cell is itself subsolution for a (single phase) "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.2448","kind":"arxiv","version":1},"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"}