{"paper":{"title":"Shape derivative approach to fractional overdetermined problems","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Ignace A. Minlend, Sidy M. Djitte","submitted_at":"2025-05-30T22:06:21Z","abstract_excerpt":"We use shape derivative approach to prove that balls are the only convex and $C^{1,1}$ regular domains in which the fractional overdetermined problem \\begin{equation*} \\left\\{\\begin{aligned} \\Ds u&= \\lambda_{s, p} u^{p-1}\\quad\\text{in}\\quad\\Om \\\\ u &= 0\\quad \\text{in}\\quad\\R^N\\setminus \\Om\\\\ u/d^s&=C_0\\quad\\text{on\\;\\; $\\partial\\O$} \\end{aligned} \\right. \\end{equation*} admits a nontrivial solution for $p\\in [1, 2]$ and where $\\lambda_{s, p}= \\lambda_{s, p}(\\O)$ is the best constant in the family of Subcritical Sobolev inequalities. In the cases $p=1$ and $p=2$, we recover the classical symmet"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.00268","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2506.00268/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}