{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:BDDPX6XSXS7J6FEOQOUW3CQPGH","short_pith_number":"pith:BDDPX6XS","schema_version":"1.0","canonical_sha256":"08c6fbfaf2bcbe9f148e83a96d8a0f31c5d5de8bf23c4efc8773920b5fe8c0de","source":{"kind":"arxiv","id":"1904.07443","version":1},"attestation_state":"computed","paper":{"title":"Some Energy Estimates for Stable Solutions to Fractional Allen-Cahn Equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Changfeng Gui, Qinfeng Li","submitted_at":"2019-04-16T03:54:11Z","abstract_excerpt":"In this paper we study stable solutions to the fractional equation \\begin{align}\n  (-\\Delta)^s u =f(u), \\quad |u| < 1 \\quad \\mbox{in $\\mathbb{R}^d$}, \\end{align}where $0<s<1$ and $f:[-1,1] \\rightarrow \\mathbb{R}$ is a $C^{1,\\alpha}$ function for $\\alpha>\\max\\{0, 1-2s\\}$. We obtain sharp energy estimates for $0<s<1/2$ and rough energy estimates for $1/2 \\le s <1$. These lead to a different proof from literature of the fact that when $d=2, \\, 0<s<1$, entire stable solutions are $1$-D solutions.\n  The scheme used in this paper is inspired by Cinti-Serra-Valdinoci[CSV17] which deals with stable no"},"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":"1904.07443","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-04-16T03:54:11Z","cross_cats_sorted":[],"title_canon_sha256":"f453ebdf9699f4a143e6cd70c8236c296d1e3c41f05408dddde9b7c00fbfac35","abstract_canon_sha256":"c75cbde9b7767aba95f1496f7d2637818ff162ccc6bc255383dc71eaf1e63c50"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:48:26.587843Z","signature_b64":"hH6Gk3Bg14vsmu30cv6j/YGXPpXvg9up4goiHsFbIGhHWap/5QhUNcWf17h8DRNh8yI9OIdeTQNioHZFQn+UAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"08c6fbfaf2bcbe9f148e83a96d8a0f31c5d5de8bf23c4efc8773920b5fe8c0de","last_reissued_at":"2026-05-17T23:48:26.587169Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:48:26.587169Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Some Energy Estimates for Stable Solutions to Fractional Allen-Cahn Equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Changfeng Gui, Qinfeng Li","submitted_at":"2019-04-16T03:54:11Z","abstract_excerpt":"In this paper we study stable solutions to the fractional equation \\begin{align}\n  (-\\Delta)^s u =f(u), \\quad |u| < 1 \\quad \\mbox{in $\\mathbb{R}^d$}, \\end{align}where $0<s<1$ and $f:[-1,1] \\rightarrow \\mathbb{R}$ is a $C^{1,\\alpha}$ function for $\\alpha>\\max\\{0, 1-2s\\}$. We obtain sharp energy estimates for $0<s<1/2$ and rough energy estimates for $1/2 \\le s <1$. These lead to a different proof from literature of the fact that when $d=2, \\, 0<s<1$, entire stable solutions are $1$-D solutions.\n  The scheme used in this paper is inspired by Cinti-Serra-Valdinoci[CSV17] which deals with stable no"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.07443","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":"1904.07443","created_at":"2026-05-17T23:48:26.587269+00:00"},{"alias_kind":"arxiv_version","alias_value":"1904.07443v1","created_at":"2026-05-17T23:48:26.587269+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1904.07443","created_at":"2026-05-17T23:48:26.587269+00:00"},{"alias_kind":"pith_short_12","alias_value":"BDDPX6XSXS7J","created_at":"2026-05-18T12:33:12.712433+00:00"},{"alias_kind":"pith_short_16","alias_value":"BDDPX6XSXS7J6FEO","created_at":"2026-05-18T12:33:12.712433+00:00"},{"alias_kind":"pith_short_8","alias_value":"BDDPX6XS","created_at":"2026-05-18T12:33:12.712433+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/BDDPX6XSXS7J6FEOQOUW3CQPGH","json":"https://pith.science/pith/BDDPX6XSXS7J6FEOQOUW3CQPGH.json","graph_json":"https://pith.science/api/pith-number/BDDPX6XSXS7J6FEOQOUW3CQPGH/graph.json","events_json":"https://pith.science/api/pith-number/BDDPX6XSXS7J6FEOQOUW3CQPGH/events.json","paper":"https://pith.science/paper/BDDPX6XS"},"agent_actions":{"view_html":"https://pith.science/pith/BDDPX6XSXS7J6FEOQOUW3CQPGH","download_json":"https://pith.science/pith/BDDPX6XSXS7J6FEOQOUW3CQPGH.json","view_paper":"https://pith.science/paper/BDDPX6XS","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1904.07443&json=true","fetch_graph":"https://pith.science/api/pith-number/BDDPX6XSXS7J6FEOQOUW3CQPGH/graph.json","fetch_events":"https://pith.science/api/pith-number/BDDPX6XSXS7J6FEOQOUW3CQPGH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BDDPX6XSXS7J6FEOQOUW3CQPGH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BDDPX6XSXS7J6FEOQOUW3CQPGH/action/storage_attestation","attest_author":"https://pith.science/pith/BDDPX6XSXS7J6FEOQOUW3CQPGH/action/author_attestation","sign_citation":"https://pith.science/pith/BDDPX6XSXS7J6FEOQOUW3CQPGH/action/citation_signature","submit_replication":"https://pith.science/pith/BDDPX6XSXS7J6FEOQOUW3CQPGH/action/replication_record"}},"created_at":"2026-05-17T23:48:26.587269+00:00","updated_at":"2026-05-17T23:48:26.587269+00:00"}