{"paper":{"title":"Regularity for Shape Optimizers: The Nondegenerate Case","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Dennis Kriventsov, Fanghua Lin","submitted_at":"2016-09-09T00:21:20Z","abstract_excerpt":"We consider minimizers of\n  \\[\n  F(\\lambda_1(\\Omega),\\ldots,\\lambda_N(\\Omega)) + |\\Omega|,\n  \\] where $F$ is a function strictly increasing in each parameter, and $\\lambda_k(\\Omega)$ is the $k$-th Dirichlet eigenvalue of $\\Omega$. Our main result is that the reduced boundary of the minimizer is composed of $C^{1,\\alpha}$ graphs, and exhausts the topological boundary except for a set of Hausdorff dimension at most $n-3$. We also obtain a new regularity result for vector-valued Bernoulli type free boundary problems."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.02624","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"}