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This proves the De Giorgi conjecture in dimension $4$ for the half-Laplacian. Equivalently, we give a positive answer to the De Giorgi conjecture for boundary reactions in $\\mathbb R^{d+1}_+=\\mathbb R^{d+1}\\cap \\{x_{d+1}\\geq 0\\}$ when $d = 4$, by proving that all critical points of $$ \\int_{\\{x_{d+1\\geq 0}\\}} \\frac12 "},"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":"1705.02781","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-05-08T08:51:27Z","cross_cats_sorted":[],"title_canon_sha256":"832804034f5151c0f59502c2729a5658049f1f3a20c9a3fe53f309072cb0195c","abstract_canon_sha256":"798189abf4a8c3f8267f426bb5ddf02f9841e0e8df20a9e2978deaa5c716b184"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:44:53.656703Z","signature_b64":"IsYr6tdGQrR1J/N7P/sCfSzj4HimTge3yLyarmzi3rfAfvCADSJVa0H2TRXv11K3aCTVt+A2+6L3V5ykDtgeDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1ff4b626b0a77da71e007c50a6816921513356af2338659cc4eb8951161ad45d","last_reissued_at":"2026-05-18T00:44:53.656053Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:44:53.656053Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On stable solutions for boundary reactions: a De Giorgi-type result in dimension 4+1","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Alessio Figalli, Joaquim Serra","submitted_at":"2017-05-08T08:51:27Z","abstract_excerpt":"We prove that every bounded stable solution of \\[ (-\\Delta)^{1/2} u + f(u) =0 \\qquad \\mbox{in }\\mathbb R^3\\] is a 1D profile, i.e., $u(x)= \\phi(e\\cdot x)$ for some $e\\in \\mathbb S^2$, where $\\phi:\\mathbb R\\to \\mathbb R$ is a nondecreasing bounded stable solution in dimension one. This proves the De Giorgi conjecture in dimension $4$ for the half-Laplacian. Equivalently, we give a positive answer to the De Giorgi conjecture for boundary reactions in $\\mathbb R^{d+1}_+=\\mathbb R^{d+1}\\cap \\{x_{d+1}\\geq 0\\}$ when $d = 4$, by proving that all critical points of $$ \\int_{\\{x_{d+1\\geq 0}\\}} \\frac12 "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.02781","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":"1705.02781","created_at":"2026-05-18T00:44:53.656149+00:00"},{"alias_kind":"arxiv_version","alias_value":"1705.02781v1","created_at":"2026-05-18T00:44:53.656149+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1705.02781","created_at":"2026-05-18T00:44:53.656149+00:00"},{"alias_kind":"pith_short_12","alias_value":"D72LMJVQU562","created_at":"2026-05-18T12:31:10.602751+00:00"},{"alias_kind":"pith_short_16","alias_value":"D72LMJVQU562OHQA","created_at":"2026-05-18T12:31:10.602751+00:00"},{"alias_kind":"pith_short_8","alias_value":"D72LMJVQ","created_at":"2026-05-18T12:31:10.602751+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/D72LMJVQU562OHQAPRIKNALJEF","json":"https://pith.science/pith/D72LMJVQU562OHQAPRIKNALJEF.json","graph_json":"https://pith.science/api/pith-number/D72LMJVQU562OHQAPRIKNALJEF/graph.json","events_json":"https://pith.science/api/pith-number/D72LMJVQU562OHQAPRIKNALJEF/events.json","paper":"https://pith.science/paper/D72LMJVQ"},"agent_actions":{"view_html":"https://pith.science/pith/D72LMJVQU562OHQAPRIKNALJEF","download_json":"https://pith.science/pith/D72LMJVQU562OHQAPRIKNALJEF.json","view_paper":"https://pith.science/paper/D72LMJVQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1705.02781&json=true","fetch_graph":"https://pith.science/api/pith-number/D72LMJVQU562OHQAPRIKNALJEF/graph.json","fetch_events":"https://pith.science/api/pith-number/D72LMJVQU562OHQAPRIKNALJEF/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/D72LMJVQU562OHQAPRIKNALJEF/action/timestamp_anchor","attest_storage":"https://pith.science/pith/D72LMJVQU562OHQAPRIKNALJEF/action/storage_attestation","attest_author":"https://pith.science/pith/D72LMJVQU562OHQAPRIKNALJEF/action/author_attestation","sign_citation":"https://pith.science/pith/D72LMJVQU562OHQAPRIKNALJEF/action/citation_signature","submit_replication":"https://pith.science/pith/D72LMJVQU562OHQAPRIKNALJEF/action/replication_record"}},"created_at":"2026-05-18T00:44:53.656149+00:00","updated_at":"2026-05-18T00:44:53.656149+00:00"}