{"paper":{"title":"The obstacle problem for the fractional Laplacian with critical drift","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":"2016-10-13T19:02:49Z","abstract_excerpt":"We study the obstacle problem for the fractional Laplacian with drift, $\\min\\left\\{(-\\Delta)^s u + b \\cdot \\nabla u,\\,u -\\varphi\\right\\} = 0$ in $\\mathbb{R}^n$, in the critical regime $s = \\frac{1}{2}$.\n  Our main result establishes the $C^{1,\\alpha}$ regularity of the free boundary around any regular point $x_0$, with an expansion of the form \\[ u(x)-\\varphi(x) = c_0\\big((x-x_0)\\cdot e\\big)_+^{1+\\tilde\\gamma(x_0)} + o\\left(|x-x_0|^{1+\\tilde\\gamma(x_0)+\\sigma}\\right), \\] \\[ \\tilde{\\gamma}(x_0) = \\frac{1}{2}+\\frac{1}{\\pi} \\arctan (b\\cdot e), \\] where $e \\in \\mathbb{S}^{n-1}$ is the normal vecto"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.04200","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"}