{"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. $n=2,3$. Just like the local case, the concepts of orientable syste"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.1193","kind":"arxiv","version":2},"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"}