{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:USO5PHRASQ7LIT3CV6O46ECZY4","short_pith_number":"pith:USO5PHRA","schema_version":"1.0","canonical_sha256":"a49dd79e20943eb44f62af9dcf1059c72a307dd615362c03a08fa992dc1533f5","source":{"kind":"arxiv","id":"1211.2622","version":4},"attestation_state":"computed","paper":{"title":"A geometric inequality and a symmetry result for elliptic systems involving the fractional Laplacian","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Andrea Pinamonti, Serena Dipierro","submitted_at":"2012-11-12T14:07:52Z","abstract_excerpt":"We study the symmetry properties for solutions of elliptic systems of the type (-\\Delta)^{s_1} u = F_1(u, v), (-\\Delta)^{s_2} v= F_2(u, v), where $F\\in C^{1,1}_{loc}(\\R^2)$, $s_1,s_2\\in (0,1)$ and the operator $(-\\Delta)^s$ is the so-called fractional Laplacian. We obtain some Poincar\\'e-type formulas for the $\\alpha$-harmonic extension in the half-space, that we use to prove a symmetry result both for stable and for monotone solutions."},"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":"1211.2622","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-11-12T14:07:52Z","cross_cats_sorted":[],"title_canon_sha256":"8ec75a02027429f8eed7e86efe0df36a8fa13540b5be63b08a7d51330bb8f473","abstract_canon_sha256":"36eefe205a29f572752d4ef75b942619eb98c610cea7e35a5560f0c86c40bc86"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:28:05.410737Z","signature_b64":"SI70eAcZYpH/7L2PWZTvM79hMpU2NkLqfx+wcpwlM47j0QIMD4UqRGiPxtD/ajU19gR4GIz51cdaCZkQt4qaDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a49dd79e20943eb44f62af9dcf1059c72a307dd615362c03a08fa992dc1533f5","last_reissued_at":"2026-05-18T03:28:05.409502Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:28:05.409502Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A geometric inequality and a symmetry result for elliptic systems involving the fractional Laplacian","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Andrea Pinamonti, Serena Dipierro","submitted_at":"2012-11-12T14:07:52Z","abstract_excerpt":"We study the symmetry properties for solutions of elliptic systems of the type (-\\Delta)^{s_1} u = F_1(u, v), (-\\Delta)^{s_2} v= F_2(u, v), where $F\\in C^{1,1}_{loc}(\\R^2)$, $s_1,s_2\\in (0,1)$ and the operator $(-\\Delta)^s$ is the so-called fractional Laplacian. We obtain some Poincar\\'e-type formulas for the $\\alpha$-harmonic extension in the half-space, that we use to prove a symmetry result both for stable and for monotone solutions."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.2622","kind":"arxiv","version":4},"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":"1211.2622","created_at":"2026-05-18T03:28:05.409609+00:00"},{"alias_kind":"arxiv_version","alias_value":"1211.2622v4","created_at":"2026-05-18T03:28:05.409609+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.2622","created_at":"2026-05-18T03:28:05.409609+00:00"},{"alias_kind":"pith_short_12","alias_value":"USO5PHRASQ7L","created_at":"2026-05-18T12:27:23.164592+00:00"},{"alias_kind":"pith_short_16","alias_value":"USO5PHRASQ7LIT3C","created_at":"2026-05-18T12:27:23.164592+00:00"},{"alias_kind":"pith_short_8","alias_value":"USO5PHRA","created_at":"2026-05-18T12:27:23.164592+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/USO5PHRASQ7LIT3CV6O46ECZY4","json":"https://pith.science/pith/USO5PHRASQ7LIT3CV6O46ECZY4.json","graph_json":"https://pith.science/api/pith-number/USO5PHRASQ7LIT3CV6O46ECZY4/graph.json","events_json":"https://pith.science/api/pith-number/USO5PHRASQ7LIT3CV6O46ECZY4/events.json","paper":"https://pith.science/paper/USO5PHRA"},"agent_actions":{"view_html":"https://pith.science/pith/USO5PHRASQ7LIT3CV6O46ECZY4","download_json":"https://pith.science/pith/USO5PHRASQ7LIT3CV6O46ECZY4.json","view_paper":"https://pith.science/paper/USO5PHRA","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1211.2622&json=true","fetch_graph":"https://pith.science/api/pith-number/USO5PHRASQ7LIT3CV6O46ECZY4/graph.json","fetch_events":"https://pith.science/api/pith-number/USO5PHRASQ7LIT3CV6O46ECZY4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/USO5PHRASQ7LIT3CV6O46ECZY4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/USO5PHRASQ7LIT3CV6O46ECZY4/action/storage_attestation","attest_author":"https://pith.science/pith/USO5PHRASQ7LIT3CV6O46ECZY4/action/author_attestation","sign_citation":"https://pith.science/pith/USO5PHRASQ7LIT3CV6O46ECZY4/action/citation_signature","submit_replication":"https://pith.science/pith/USO5PHRASQ7LIT3CV6O46ECZY4/action/replication_record"}},"created_at":"2026-05-18T03:28:05.409609+00:00","updated_at":"2026-05-18T03:28:05.409609+00:00"}