{"paper":{"title":"Rigidity results for non local phase transitions in the Heisenberg group $H$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.AP","authors_text":"Luis F. L\\'opez, Yannick Sire","submitted_at":"2013-06-14T16:03:26Z","abstract_excerpt":"In the Heisenberg group framework, we study rigidity properties for stable solutions of $(-\\Delta_H)^s v = f(v)$ in $H$, $s \\in (0,1)$. We obtain a Poincar\\'e type inequality in connection with a degenerate elliptic equation in $\\R^4_+$; through an extension (or \"lifting\") procedure, this inequality will be then used for giving a criterion under which the level sets of the above solutions are minimal surfaces in $H$, i.e. they have vanishing mean curvature."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.3438","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"}