{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:576HOQ7MIAPGZHYJUDPAEQU2L6","short_pith_number":"pith:576HOQ7M","schema_version":"1.0","canonical_sha256":"effc7743ec401e6c9f09a0de02429a5f92de569da5ed1d25b438a28638a8c6c2","source":{"kind":"arxiv","id":"2605.23666","version":1},"attestation_state":"computed","paper":{"title":"A Priori Regularity Estimates for Ratio of Solutions to Elliptic Equations with a Product Structure of Two-Dimensional Nodal Sets","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Gabriele Fioravanti","submitted_at":"2026-05-22T14:14:43Z","abstract_excerpt":"In this paper, we establish optimal a priori $C^{1,\\alpha}$ regularity estimates for the ratio $w = v/u$ of two solutions to the same elliptic equation $-\\operatorname{div}(A \\nabla u )=0$ with Lipschitz coefficients $A$, under the assumption that their nodal sets satisfy $Z(u) \\subseteq Z(v)$. We specifically address the case where the zero set $Z(u)$ exhibits a product structure of $2$-dimensional nodal sets, namely $Z(u)=Z(u_1)\\times \\cdots \\times Z(u_{m})$, where the $u_i$ are $2$-dimensional functions. This result extends the regularity estimates previously proved in dimension $2$ by [Log"},"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":"2605.23666","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-05-22T14:14:43Z","cross_cats_sorted":[],"title_canon_sha256":"67537caf04a2baec46eb1dfe78f5e2f9fd5fc78bdce583aa6defe62ad81a9e21","abstract_canon_sha256":"aa96dbcd37187e9d0abd4f54eabfea91c3cb84d0e2139c920cf7855d0565109a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-25T02:02:24.995599Z","signature_b64":"AXL2EpPTeAFdplNmXfASwxu0Q1hmL+TpYLInFz2IqVycfqc9o/epLLzLVRMNaYoMb+zWF6O71nQUZG8BVUZZBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"effc7743ec401e6c9f09a0de02429a5f92de569da5ed1d25b438a28638a8c6c2","last_reissued_at":"2026-05-25T02:02:24.994723Z","signature_status":"signed_v1","first_computed_at":"2026-05-25T02:02:24.994723Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Priori Regularity Estimates for Ratio of Solutions to Elliptic Equations with a Product Structure of Two-Dimensional Nodal Sets","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Gabriele Fioravanti","submitted_at":"2026-05-22T14:14:43Z","abstract_excerpt":"In this paper, we establish optimal a priori $C^{1,\\alpha}$ regularity estimates for the ratio $w = v/u$ of two solutions to the same elliptic equation $-\\operatorname{div}(A \\nabla u )=0$ with Lipschitz coefficients $A$, under the assumption that their nodal sets satisfy $Z(u) \\subseteq Z(v)$. We specifically address the case where the zero set $Z(u)$ exhibits a product structure of $2$-dimensional nodal sets, namely $Z(u)=Z(u_1)\\times \\cdots \\times Z(u_{m})$, where the $u_i$ are $2$-dimensional functions. This result extends the regularity estimates previously proved in dimension $2$ by [Log"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.23666","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.23666/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2605.23666","created_at":"2026-05-25T02:02:24.994851+00:00"},{"alias_kind":"arxiv_version","alias_value":"2605.23666v1","created_at":"2026-05-25T02:02:24.994851+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.23666","created_at":"2026-05-25T02:02:24.994851+00:00"},{"alias_kind":"pith_short_12","alias_value":"576HOQ7MIAPG","created_at":"2026-05-25T02:02:24.994851+00:00"},{"alias_kind":"pith_short_16","alias_value":"576HOQ7MIAPGZHYJ","created_at":"2026-05-25T02:02:24.994851+00:00"},{"alias_kind":"pith_short_8","alias_value":"576HOQ7M","created_at":"2026-05-25T02:02:24.994851+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/576HOQ7MIAPGZHYJUDPAEQU2L6","json":"https://pith.science/pith/576HOQ7MIAPGZHYJUDPAEQU2L6.json","graph_json":"https://pith.science/api/pith-number/576HOQ7MIAPGZHYJUDPAEQU2L6/graph.json","events_json":"https://pith.science/api/pith-number/576HOQ7MIAPGZHYJUDPAEQU2L6/events.json","paper":"https://pith.science/paper/576HOQ7M"},"agent_actions":{"view_html":"https://pith.science/pith/576HOQ7MIAPGZHYJUDPAEQU2L6","download_json":"https://pith.science/pith/576HOQ7MIAPGZHYJUDPAEQU2L6.json","view_paper":"https://pith.science/paper/576HOQ7M","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2605.23666&json=true","fetch_graph":"https://pith.science/api/pith-number/576HOQ7MIAPGZHYJUDPAEQU2L6/graph.json","fetch_events":"https://pith.science/api/pith-number/576HOQ7MIAPGZHYJUDPAEQU2L6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/576HOQ7MIAPGZHYJUDPAEQU2L6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/576HOQ7MIAPGZHYJUDPAEQU2L6/action/storage_attestation","attest_author":"https://pith.science/pith/576HOQ7MIAPGZHYJUDPAEQU2L6/action/author_attestation","sign_citation":"https://pith.science/pith/576HOQ7MIAPGZHYJUDPAEQU2L6/action/citation_signature","submit_replication":"https://pith.science/pith/576HOQ7MIAPGZHYJUDPAEQU2L6/action/replication_record"}},"created_at":"2026-05-25T02:02:24.994851+00:00","updated_at":"2026-05-25T02:02:24.994851+00:00"}