{"paper":{"title":"The energy of fractional Allen--Cahn layers in dimension one","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Alvaro Carballeira, Damien Galant, Javier G\\'omez-Serrano","submitted_at":"2026-08-06T16:15:18Z","abstract_excerpt":"We study the energy $\\mathcal{E}(s) := E_{s}[\\Phi_s]$ of the one-dimensional fractional Allen--Cahn layer solution $\\Phi_s$, defined as the unique odd, increasing solution of $(-\\Delta)^{s} \\Phi_s = \\Phi_s - \\Phi_s^{3}$ with $\\Phi_s(\\pm\\infty)=\\pm 1$ and $\\Phi_s(0)=0$, for $s\\in(1/2,1]$. Our main results are a sharp qualitative and quantitative description of the energy $\\mathcal{E}$ on this interval. We show that the energy is continuous and strictly decreasing, and we obtain explicit asymptotic expansions at both endpoints. At the upper endpoint we prove $\\mathcal{E}(s) = \\frac{2\\sqrt{2}}{3}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.06226","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2608.06226/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"}