{"paper":{"title":"Uniqueness of the minimizer for a random nonlocal functional with double-well potential in $d\\le2$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.MP"],"primary_cat":"math-ph","authors_text":"Enza Orlandi, Nicolas Dir","submitted_at":"2013-08-25T09:35:08Z","abstract_excerpt":"We consider a small random perturbation of the energy functional $$ [u]^2_{H^s(\\Lambda, R^d)} + \\int_\\Lambda W(u(x)) dx $$ for $s \\in (0,1),$ where the non-local part $ [u]^2_{H^s(\\Lambda,R^d)}$ denotes the total contribution from $\\Lambda \\subset R^d$ in the $H^s (R^d)$ Gagliardo semi-norm of $u$ and $W$ is a double well potential. We show that there exists, as $\\Lambda $ invades $ R^d$, for almost all realizations of the random term a minimizer under compact perturbations, which is unique when $d=2$, $s \\in (\\frac 12,1)$ and when $d=1$, $s \\in [\\frac 14, 1).$ This uniqueness is a consequence"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.5391","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"}