{"paper":{"title":"A geometric inequality and a symmetry result for elliptic systems involving the fractional Laplacian","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Andrea Pinamonti, Serena Dipierro","submitted_at":"2012-11-12T14:07:52Z","abstract_excerpt":"We study the symmetry properties for solutions of elliptic systems of the type (-\\Delta)^{s_1} u = F_1(u, v), (-\\Delta)^{s_2} v= F_2(u, v), where $F\\in C^{1,1}_{loc}(\\R^2)$, $s_1,s_2\\in (0,1)$ and the operator $(-\\Delta)^s$ is the so-called fractional Laplacian. We obtain some Poincar\\'e-type formulas for the $\\alpha$-harmonic extension in the half-space, that we use to prove a symmetry result both for stable and for monotone solutions."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.2622","kind":"arxiv","version":4},"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"}