{"paper":{"title":"Local and global minimizers for a variational energy involving a fractional norm","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Enrico Valdinoci, Giampiero Palatucci, Ovidiu Savin","submitted_at":"2011-04-09T17:15:12Z","abstract_excerpt":"We study existence, unicity and other geometric properties of the minimizers of the energy functional $$ \\|u\\|^2_{H^s(\\Omega)}+\\int_\\Omega W(u)\\,dx, $$ 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. We also deal with the solutions of the related fractional elliptic Allen-Cahn equation on the entire space $\\mathbb{R}^n$.\n  The results collected here will also be useful for forthcoming papers, where the second and the third author will study the $\\Gamma$-convergence and the density estimates for level sets of "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1104.1725","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":""},"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"}