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In particular, if $\\Omega$ is a ball $B \\subset \\mathbb{R}^N$, we obtain the following chain of inequalities: $$ \\lambda_2(p;B) \\leq \\dots \\leq \\lambda_{N+1}(p;B) \\leq \\lambda_\\ominus(p;B). $$ Here $\\lambda_i(p;B)"},"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":"1704.03194","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-04-11T08:33:58Z","cross_cats_sorted":["math.SP"],"title_canon_sha256":"3a58e185b2f5a056c71b560e3573689616795436e4caf39fa9c0dcbf27cb0619","abstract_canon_sha256":"3a5efd3ab13074d7620f14c9f18fd85ba1bbe9f40445a35c5c30979bf43c8d70"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:01:10.552831Z","signature_b64":"k0I7Vti4qV3O8ZDIdC4usO9vEvG8xCpwNiqcRAsrYukNRy0F0FEYtfEi8yMGEnDUAR7gmWx7jPXX9c2QAudlBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"04448fd016a753e3e792a4247e374694dc6e08a8884256f17429258777df4350","last_reissued_at":"2026-05-18T00:01:10.552284Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:01:10.552284Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On multiplicity of eigenvalues and symmetry of eigenfunctions of the $p$-Laplacian","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.SP"],"primary_cat":"math.AP","authors_text":"Benjamin Audoux, Enea Parini, Vladimir Bobkov","submitted_at":"2017-04-11T08:33:58Z","abstract_excerpt":"We investigate multiplicity and symmetry properties of higher eigenvalues and eigenfunctions of the $p$-Laplacian under homogeneous Dirichlet boundary conditions on certain symmetric domains $\\Omega \\subset \\mathbb{R}^N$. 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In particular, if $\\Omega$ is a ball $B \\subset \\mathbb{R}^N$, we obtain the following chain of inequalities: $$ \\lambda_2(p;B) \\leq \\dots \\leq \\lambda_{N+1}(p;B) \\leq \\lambda_\\ominus(p;B). $$ Here $\\lambda_i(p;B)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1704.03194","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":"1704.03194","created_at":"2026-05-18T00:01:10.552385+00:00"},{"alias_kind":"arxiv_version","alias_value":"1704.03194v1","created_at":"2026-05-18T00:01:10.552385+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1704.03194","created_at":"2026-05-18T00:01:10.552385+00:00"},{"alias_kind":"pith_short_12","alias_value":"ARCI7UAWU5J6","created_at":"2026-05-18T12:31:05.417338+00:00"},{"alias_kind":"pith_short_16","alias_value":"ARCI7UAWU5J6HZ4S","created_at":"2026-05-18T12:31:05.417338+00:00"},{"alias_kind":"pith_short_8","alias_value":"ARCI7UAW","created_at":"2026-05-18T12:31:05.417338+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/ARCI7UAWU5J6HZ4SUQSH4N2GST","json":"https://pith.science/pith/ARCI7UAWU5J6HZ4SUQSH4N2GST.json","graph_json":"https://pith.science/api/pith-number/ARCI7UAWU5J6HZ4SUQSH4N2GST/graph.json","events_json":"https://pith.science/api/pith-number/ARCI7UAWU5J6HZ4SUQSH4N2GST/events.json","paper":"https://pith.science/paper/ARCI7UAW"},"agent_actions":{"view_html":"https://pith.science/pith/ARCI7UAWU5J6HZ4SUQSH4N2GST","download_json":"https://pith.science/pith/ARCI7UAWU5J6HZ4SUQSH4N2GST.json","view_paper":"https://pith.science/paper/ARCI7UAW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1704.03194&json=true","fetch_graph":"https://pith.science/api/pith-number/ARCI7UAWU5J6HZ4SUQSH4N2GST/graph.json","fetch_events":"https://pith.science/api/pith-number/ARCI7UAWU5J6HZ4SUQSH4N2GST/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ARCI7UAWU5J6HZ4SUQSH4N2GST/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ARCI7UAWU5J6HZ4SUQSH4N2GST/action/storage_attestation","attest_author":"https://pith.science/pith/ARCI7UAWU5J6HZ4SUQSH4N2GST/action/author_attestation","sign_citation":"https://pith.science/pith/ARCI7UAWU5J6HZ4SUQSH4N2GST/action/citation_signature","submit_replication":"https://pith.science/pith/ARCI7UAWU5J6HZ4SUQSH4N2GST/action/replication_record"}},"created_at":"2026-05-18T00:01:10.552385+00:00","updated_at":"2026-05-18T00:01:10.552385+00:00"}