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We prove De Giorgi type results for this system for certain values of $\\mathbf s$ and in lower dimensions, i.e. $n=2,3$. Just like the local case, the concepts of orientable syste"},"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":"1402.1193","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-02-05T21:41:50Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"1bc59cded45766f1962dbcac9aa9418da56e833f1a1106d26bc69b6ac2c56a00","abstract_canon_sha256":"01776e6f2df470c50e7bc2983689698038fb00952dc155d9b2fd351ab3847908"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:27:05.767675Z","signature_b64":"xh8+3P3VQ7O2/9lSfvEoSeXyEEXXBL5eFV+Pc2xdayntx/sMYtWYBBWsGrzYsWZYxDPoijTiiYWNPhKrWXtDCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7725d3a71a7a3d80cfb6589a5da3de759da29749bba7f4df1947692e79c77e27","last_reissued_at":"2026-05-18T01:27:05.767137Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:27:05.767137Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Symmetry results for fractional elliptic systems and related problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.AP","authors_text":"Mostafa Fazly, Yannick Sire","submitted_at":"2014-02-05T21:41:50Z","abstract_excerpt":"We study elliptic gradient systems with fractional laplacian operators on the whole space $$ (- \\Delta)^\\mathbf s \\mathbf u =\\nabla H (\\mathbf u) \\ \\ \\text{in}\\ \\ \\mathbf{R}^n,$$ where $\\mathbf u:\\mathbf{R}^n\\to \\mathbf{R}^m$, $H\\in C^{2,\\gamma}(\\mathbf{R}^m)$ for $\\gamma > \\max(0,1-2\\min \\left \\{s_i \\right \\})$, $\\mathbf s=(s_1,\\cdots,s_m)$ for $0<s_i<1$ and $\\nabla H (\\mathbf u)=(H_{u_i}(u_1, u_2,\\cdots,u_m))_{i}$. We prove De Giorgi type results for this system for certain values of $\\mathbf s$ and in lower dimensions, i.e. $n=2,3$. 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