{"paper":{"title":"Nonlocal $s$-minimal surfaces and Lawson cones","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.AP","authors_text":"Juan D\\'avila, Juncheng Wei, Manuel del Pino","submitted_at":"2014-02-17T22:55:55Z","abstract_excerpt":"The nonlocal $s$-fractional minimal surface equation for $\\Sigma= \\partial E$ where $E$ is an open set in $R^N$ is given by $$ H_\\Sigma^ s (p) := \\int_{R^N} \\frac {\\chi_E(x) - \\chi_{E^c}(x)} {|x-p|^{N+s}}\\, dx \\ =\\ 0 \\quad \\text{for all } p\\in \\Sigma. $$ Here $0<s<1$, $\\chi$ designates characteristic function, and the integral is understood in the principal value sense. The classical notion of minimal surface is recovered by letting $s\\to 1$. In this paper we exhibit the first concrete examples (beyond the plane) of nonlocal $s-$minimal surfaces. When $s$ is close to $1$, we first construct a "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.4173","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"}