{"paper":{"title":"$\\Gamma$-convergence for nonlocal phase transitions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Enrico Valdinoci, Ovidiu Savin","submitted_at":"2010-07-10T14:55:51Z","abstract_excerpt":"We discuss the $\\Gamma$-convergence, under the appropriate scaling, of the energy functional $$ \\|u\\|_{H^s(\\Omega)}^2+\\int_\\Omega W(u)dx,$$ with $s \\in (0,1)$, where $\\|u\\|_{H^s(\\Omega)}$ denotes the total contribution from $\\Omega$ in the $H^s$ norm of $u$, and $W$ is a double-well potential.\n  When $s\\in [1/2,\\,1)$, we show that the energy $\\Gamma$-converges to the classical minimal surface functional -- while, when $s\\in(0,\\,1/2)$, it is easy to see that the functional $\\Gamma$-converges to the nonlocal minimal surface functional."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1007.1725","kind":"arxiv","version":3},"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"}