{"paper":{"title":"On global solutions to semilinear elliptic equations related to the one-phase free boundary problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Xavier Fern\\'andez-Real, Xavier Ros-Oton","submitted_at":"2018-11-07T16:46:59Z","abstract_excerpt":"Motivated by its relation to models of flame propagation, we study globally Lipschitz solutions of $\\Delta u=f(u)$ in $\\mathbb{R}^n$, where $f$ is smooth, non-negative, with support in the interval $[0,1]$. In such setting, any \"blow-down\" of the solution $u$ will converge to a global solution to the classical one-phase free boundary problem of Alt-Caffarelli.\n  In analogy to a famous theorem of Savin for the Allen-Cahn equation, we study here the 1D symmetry of solutions $u$ that are energy minimizers. Our main result establishes that, in dimensions $n<6$, if $u$ is axially symmetric and stab"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.02980","kind":"arxiv","version":1},"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"}