{"paper":{"title":"Regularity of the optimal shape for the first eigenvalue of the Laplacian with volume and inclusion constraints","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Jimmy Lamboley (IRMAR), Tanguy Brian\\c{c}on (IRMAR)","submitted_at":"2008-07-14T16:32:53Z","abstract_excerpt":"We consider the well-known following shape optimization problem: $$\\lambda_1(\\Omega^*)=\\min_{\\stackrel{|\\Omega|=a} {\\Omega\\subset{D}}} \\lambda_1(\\Omega), $$ where $\\lambda_1$ denotes the first eigenvalue of the Laplace operator with homogeneous Dirichlet boundary condition, and $D$ is an open bounded set (a box). It is well-known that the solution of this problem is the ball of volume $a$ if such a ball exists in the box $D$ (Faber-Krahn's theorem). In this paper, we prove regularity properties of the boundary of the optimal shapes $\\Omega^*$ in any case and in any dimension. Full regularity i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0807.2196","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"}