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Just like in the proofs of the classical De Giorgi's conjecture in dimension 2 (Ghoussoub-Gui) and in dimension 3 (Ambrosio-Cabr\\'{e}), the key step is a Liouville theorem for linear systems. We also give an"},"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":"1203.6114","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-03-27T23:43:40Z","cross_cats_sorted":[],"title_canon_sha256":"1fd197528da17e499c108d3a6bcff603c7653f6d3e7d5cea532360935f829f5e","abstract_canon_sha256":"e3a21da28fe91747a090afddfa62a76055db89bc1efa332e34dd74006660a7e3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:57:15.679166Z","signature_b64":"Bn5cppl2W7LbKPa7G675fHeWQSydS3pdKq+frNMcx/Dz/HbtfK6XrjBh8tOfbnLoBO9xJNDw3OjcG3GS8+NlAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"459da9b34b30f8df640c075e5a93e203b8e1e1ab5cb3192e8088b5013fd6fc6d","last_reissued_at":"2026-05-18T03:57:15.678649Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:57:15.678649Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"De Giorgi type results for elliptic systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Mostafa Fazly, Nassif Ghoussoub","submitted_at":"2012-03-27T23:43:40Z","abstract_excerpt":"We consider the following elliptic system\n  \\Delta u =\\nabla H (u) \\ \\ \\text{in}\\ \\ \\mathbf{R}^N,\nwhere $u:\\mathbf{R}^N\\to \\mathbf{R}^m$ and $H\\in C^2(\\mathbf{R}^m)$, and prove, under various conditions on the nonlinearity $H$ that, at least in low dimensions, a solution $u=(u_i)_{i=1}^m$ is necessarily one-dimensional whenever each one of its components $u_i$ is monotone in one direction. Just like in the proofs of the classical De Giorgi's conjecture in dimension 2 (Ghoussoub-Gui) and in dimension 3 (Ambrosio-Cabr\\'{e}), the key step is a Liouville theorem for linear systems. 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