{"paper":{"title":"Characterization of $f$-extremal disks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.AP","authors_text":"Jos\\'e M. Espinar, Laurent Mazet","submitted_at":"2016-10-31T09:04:47Z","abstract_excerpt":"We show uniqueness for overdetermined elliptic problems defined on topological disks $\\Omega$ with $C^2$ boundary, i.e., positive solutions $u$ to $\\Delta u + f(u)=0$ in $\\Omega \\subset (M^2,g)$ so that $u = 0$ and $\\frac{\\partial u}{\\partial \\vec\\eta} = cte $ along $\\partial \\Omega$, $\\vec\\eta$ the unit outward normal along $\\partial\\Omega$ under the assumption of the existence of a candidate family. To do so, we adapt the G\\'alvez-Mira generalized Hopf-type Theorem to the realm of overdetermined elliptic problem.\n  When $(M^2,g)$ is the standard sphere $\\mathbb S^2$ and $f$ is a $C^1$ functi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.09833","kind":"arxiv","version":3},"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"}