{"paper":{"title":"An $L^\\infty$-variational problem involving the Fractional Laplacian","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Roger Moser, Simone Carano","submitted_at":"2025-10-16T09:20:37Z","abstract_excerpt":"For $s\\in(0,1)$ and an open bounded set $\\Omega\\subset\\mathbb R^n$, we prove existence and uniqueness of absolute minimisers of the supremal functional $$E_\\infty(u)=\\|(-\\Delta)^s u\\|_{L^\\infty(\\mathbb R^n)},$$ where $(-\\Delta)^s$ is the Fractional Laplacian of order $s$ and $u$ has prescribed Dirichlet data in the complement of $\\Omega$. We further show that the minimiser $u_\\infty$ satisfies the (fractional) PDE $$ (-\\Delta)^s u_\\infty=E_\\infty(u_\\infty)\\,\\mathrm{sgn}f_\\infty \\qquad\\mbox{in }\\Omega, $$ for some analytic function $f_\\infty\\in L^1(\\Omega)$ obtained as the restriction of an $s$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2510.14476","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/2510.14476/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"}