{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2010:ZYT2JQDA7ANKHWAS6IIDASOVQF","short_pith_number":"pith:ZYT2JQDA","schema_version":"1.0","canonical_sha256":"ce27a4c060f81aa3d812f2103049d5815ed6452722b99d5fcd041febf7ea5bf8","source":{"kind":"arxiv","id":"1010.6239","version":1},"attestation_state":"computed","paper":{"title":"About H\\\"older-regularity of the convex shape minimizing {\\lambda}2","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Jimmy Lamboley (CEREMADE)","submitted_at":"2010-10-29T15:04:06Z","abstract_excerpt":"In this paper, we consider the well-known following shape optimization problem: $$\\lambda_2(\\Omega^*)=\\min_{\\stackrel{|\\Omega|=V_0} {\\Omega\\textrm{ convex}}} \\lambda_2(\\Omega),$$ where $\\lambda_2(\\Om)$ denotes the second eigenvalue of the Laplace operator with homogeneous Dirichlet boundary conditions in $\\Om\\subset\\R^2$, and $|\\Om|$ is the area of $\\Om$. We prove, under some technical assumptions, that any optimal shape $\\Omega^*$ is $\\mathcal{C}^{1,\\frac{1}{2}}$ and is not $\\C^{1,\\alpha}$ for any $\\alpha>\\frac{1}{2}$. We also derive from our strategy some more general regularity results, in "},"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":"1010.6239","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2010-10-29T15:04:06Z","cross_cats_sorted":[],"title_canon_sha256":"7031f0e5f763a1f83f98abac22e7488e0111868c593917097c1a38bef2c4e7d1","abstract_canon_sha256":"4add1193b77cf9297c7f30ba22179fb88a34afa0ced0141fd1c34088b2261385"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:38:07.263884Z","signature_b64":"VHjIwom8jns7eZK33+A31ieAu5OAguF849SZCO7igz2ojaw65Bh9bBZDByWTJyStklY59ecXjcO1IMtmjDbSAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ce27a4c060f81aa3d812f2103049d5815ed6452722b99d5fcd041febf7ea5bf8","last_reissued_at":"2026-05-18T04:38:07.263430Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:38:07.263430Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"About H\\\"older-regularity of the convex shape minimizing {\\lambda}2","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Jimmy Lamboley (CEREMADE)","submitted_at":"2010-10-29T15:04:06Z","abstract_excerpt":"In this paper, we consider the well-known following shape optimization problem: $$\\lambda_2(\\Omega^*)=\\min_{\\stackrel{|\\Omega|=V_0} {\\Omega\\textrm{ convex}}} \\lambda_2(\\Omega),$$ where $\\lambda_2(\\Om)$ denotes the second eigenvalue of the Laplace operator with homogeneous Dirichlet boundary conditions in $\\Om\\subset\\R^2$, and $|\\Om|$ is the area of $\\Om$. We prove, under some technical assumptions, that any optimal shape $\\Omega^*$ is $\\mathcal{C}^{1,\\frac{1}{2}}$ and is not $\\C^{1,\\alpha}$ for any $\\alpha>\\frac{1}{2}$. We also derive from our strategy some more general regularity results, in "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1010.6239","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1010.6239","created_at":"2026-05-18T04:38:07.263496+00:00"},{"alias_kind":"arxiv_version","alias_value":"1010.6239v1","created_at":"2026-05-18T04:38:07.263496+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1010.6239","created_at":"2026-05-18T04:38:07.263496+00:00"},{"alias_kind":"pith_short_12","alias_value":"ZYT2JQDA7ANK","created_at":"2026-05-18T12:26:18.847500+00:00"},{"alias_kind":"pith_short_16","alias_value":"ZYT2JQDA7ANKHWAS","created_at":"2026-05-18T12:26:18.847500+00:00"},{"alias_kind":"pith_short_8","alias_value":"ZYT2JQDA","created_at":"2026-05-18T12:26:18.847500+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/ZYT2JQDA7ANKHWAS6IIDASOVQF","json":"https://pith.science/pith/ZYT2JQDA7ANKHWAS6IIDASOVQF.json","graph_json":"https://pith.science/api/pith-number/ZYT2JQDA7ANKHWAS6IIDASOVQF/graph.json","events_json":"https://pith.science/api/pith-number/ZYT2JQDA7ANKHWAS6IIDASOVQF/events.json","paper":"https://pith.science/paper/ZYT2JQDA"},"agent_actions":{"view_html":"https://pith.science/pith/ZYT2JQDA7ANKHWAS6IIDASOVQF","download_json":"https://pith.science/pith/ZYT2JQDA7ANKHWAS6IIDASOVQF.json","view_paper":"https://pith.science/paper/ZYT2JQDA","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1010.6239&json=true","fetch_graph":"https://pith.science/api/pith-number/ZYT2JQDA7ANKHWAS6IIDASOVQF/graph.json","fetch_events":"https://pith.science/api/pith-number/ZYT2JQDA7ANKHWAS6IIDASOVQF/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ZYT2JQDA7ANKHWAS6IIDASOVQF/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ZYT2JQDA7ANKHWAS6IIDASOVQF/action/storage_attestation","attest_author":"https://pith.science/pith/ZYT2JQDA7ANKHWAS6IIDASOVQF/action/author_attestation","sign_citation":"https://pith.science/pith/ZYT2JQDA7ANKHWAS6IIDASOVQF/action/citation_signature","submit_replication":"https://pith.science/pith/ZYT2JQDA7ANKHWAS6IIDASOVQF/action/replication_record"}},"created_at":"2026-05-18T04:38:07.263496+00:00","updated_at":"2026-05-18T04:38:07.263496+00:00"}