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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."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1609.02624","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-09-09T00:21:20Z","cross_cats_sorted":[],"title_canon_sha256":"14d00f6ef2ddc54707a6d28bfa59ae8b8dd4addcd3bba816d436273fa76fc5c0","abstract_canon_sha256":"41a9d9c9975f6b45bd8b9504906a8c340a6c9d66c89beb08140d6bac18de412c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:42:17.723675Z","signature_b64":"1gL5QOVGAkP5vzZy3xhD3Fbv9kqqyuWM1hZIyYZ1kB43/p51ybOceWKNL73VQpgxz0xNqT2hDq7Ht0vzk9aGBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5a482267189904d189043a25ad467bf5441c552bd7af6bc4ef690cf6d71b4e47","last_reissued_at":"2026-05-18T00:42:17.723182Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:42:17.723182Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1609.02624","created_at":"2026-05-18T00:42:17.723242+00:00"},{"alias_kind":"arxiv_version","alias_value":"1609.02624v3","created_at":"2026-05-18T00:42:17.723242+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.02624","created_at":"2026-05-18T00:42:17.723242+00:00"},{"alias_kind":"pith_short_12","alias_value":"LJECEZYYTECN","created_at":"2026-05-18T12:30:29.479603+00:00"},{"alias_kind":"pith_short_16","alias_value":"LJECEZYYTECNDCIE","created_at":"2026-05-18T12:30:29.479603+00:00"},{"alias_kind":"pith_short_8","alias_value":"LJECEZYY","created_at":"2026-05-18T12:30:29.479603+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/LJECEZYYTECNDCIEHIS22RT36V","json":"https://pith.science/pith/LJECEZYYTECNDCIEHIS22RT36V.json","graph_json":"https://pith.science/api/pith-number/LJECEZYYTECNDCIEHIS22RT36V/graph.json","events_json":"https://pith.science/api/pith-number/LJECEZYYTECNDCIEHIS22RT36V/events.json","paper":"https://pith.science/paper/LJECEZYY"},"agent_actions":{"view_html":"https://pith.science/pith/LJECEZYYTECNDCIEHIS22RT36V","download_json":"https://pith.science/pith/LJECEZYYTECNDCIEHIS22RT36V.json","view_paper":"https://pith.science/paper/LJECEZYY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1609.02624&json=true","fetch_graph":"https://pith.science/api/pith-number/LJECEZYYTECNDCIEHIS22RT36V/graph.json","fetch_events":"https://pith.science/api/pith-number/LJECEZYYTECNDCIEHIS22RT36V/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LJECEZYYTECNDCIEHIS22RT36V/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LJECEZYYTECNDCIEHIS22RT36V/action/storage_attestation","attest_author":"https://pith.science/pith/LJECEZYYTECNDCIEHIS22RT36V/action/author_attestation","sign_citation":"https://pith.science/pith/LJECEZYYTECNDCIEHIS22RT36V/action/citation_signature","submit_replication":"https://pith.science/pith/LJECEZYYTECNDCIEHIS22RT36V/action/replication_record"}},"created_at":"2026-05-18T00:42:17.723242+00:00","updated_at":"2026-05-18T00:42:17.723242+00:00"}