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We consider the harmonic map equation $$ -\\Delta u=|\\nabla u|^2u,$$ subject to the Dirichlet boundary condition $ u(e^{i\\theta})=(R\\cos\\theta,R\\sin\\theta,\\sqrt{1-R^2}):=g_R$, where $0<R<1$ and $u: B_1\\to {\\mathbb S}^2$ is understood in the weak harmonic-map sense. In 1983, Brezis and Coron proved the existence of two explicit solutions of this nonlinear Dirichlet\n  problem and showed that they are the unique minimizers in their respective relative homotopy classes.\n  In this paper, we resolve a long-standing open question originally posed in their"},"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":"2606.08648","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-06-07T14:25:18Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"972a095a570af42aa4a4d5aee1e93eda79ab195cdb24e27d3cd0000255043968","abstract_canon_sha256":"53e6bad0b09b14a613d275ee6e4dbba998f20ec5e2f591483568f29d0e28458f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-09T01:05:42.655530Z","signature_b64":"nu0dKYf8gkfeslnyK5n+50/8S0zvQR8DQNDcWH68tAmCJN1AUsHasRPMV8ZCPC0YPTnjHxTFSPuOODol5s+HBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5c6bb3867c30e319f54be51bbf28eb0735e40ee3a3061bb32d2fd9bcfba8d585","last_reissued_at":"2026-06-09T01:05:42.655088Z","signature_status":"signed_v1","first_computed_at":"2026-06-09T01:05:42.655088Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On Brezis Open Problem 3.1","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.AP","authors_text":"Fanghua Lin, Juncheng Wei, Qi Guo, Xueping Huang, Yi C. 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In 1983, Brezis and Coron proved the existence of two explicit solutions of this nonlinear Dirichlet\n  problem and showed that they are the unique minimizers in their respective relative homotopy classes.\n  In this paper, we resolve a long-standing open question originally posed in their"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.08648","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/2606.08648/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":"2606.08648","created_at":"2026-06-09T01:05:42.655156+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.08648v1","created_at":"2026-06-09T01:05:42.655156+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.08648","created_at":"2026-06-09T01:05:42.655156+00:00"},{"alias_kind":"pith_short_12","alias_value":"LRV3HBT4GDRR","created_at":"2026-06-09T01:05:42.655156+00:00"},{"alias_kind":"pith_short_16","alias_value":"LRV3HBT4GDRRT5KL","created_at":"2026-06-09T01:05:42.655156+00:00"},{"alias_kind":"pith_short_8","alias_value":"LRV3HBT4","created_at":"2026-06-09T01:05:42.655156+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/LRV3HBT4GDRRT5KL4UN36KHLA4","json":"https://pith.science/pith/LRV3HBT4GDRRT5KL4UN36KHLA4.json","graph_json":"https://pith.science/api/pith-number/LRV3HBT4GDRRT5KL4UN36KHLA4/graph.json","events_json":"https://pith.science/api/pith-number/LRV3HBT4GDRRT5KL4UN36KHLA4/events.json","paper":"https://pith.science/paper/LRV3HBT4"},"agent_actions":{"view_html":"https://pith.science/pith/LRV3HBT4GDRRT5KL4UN36KHLA4","download_json":"https://pith.science/pith/LRV3HBT4GDRRT5KL4UN36KHLA4.json","view_paper":"https://pith.science/paper/LRV3HBT4","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.08648&json=true","fetch_graph":"https://pith.science/api/pith-number/LRV3HBT4GDRRT5KL4UN36KHLA4/graph.json","fetch_events":"https://pith.science/api/pith-number/LRV3HBT4GDRRT5KL4UN36KHLA4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LRV3HBT4GDRRT5KL4UN36KHLA4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LRV3HBT4GDRRT5KL4UN36KHLA4/action/storage_attestation","attest_author":"https://pith.science/pith/LRV3HBT4GDRRT5KL4UN36KHLA4/action/author_attestation","sign_citation":"https://pith.science/pith/LRV3HBT4GDRRT5KL4UN36KHLA4/action/citation_signature","submit_replication":"https://pith.science/pith/LRV3HBT4GDRRT5KL4UN36KHLA4/action/replication_record"}},"created_at":"2026-06-09T01:05:42.655156+00:00","updated_at":"2026-06-09T01:05:42.655156+00:00"}