{"paper":{"title":"One-dimensional solutions of non-local Allen-Cahn-type equations with rough kernels","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Matteo Cozzi, Tommaso Passalacqua","submitted_at":"2015-10-09T20:19:23Z","abstract_excerpt":"We are interested in the study of local and global minimizers for an energy functional of the type $$ \\frac{1}{4} \\iint_{\\mathbb{R}^{2 N} \\setminus \\left( \\mathbb{R}^N \\setminus \\Omega \\right)^2} |u(x) - u(y)|^2 K(x - y) \\, dx dy + \\int_{\\Omega} W(u(x)) \\, dx, $$ where $W$ is a smooth, even double-well potential and $K$ is a non-negative symmetric kernel in a general class, which contains as a particular case the choice $K(z) = |z|^{- N - 2 s}$, with $s \\in (0, 1)$, related to the fractional Laplacian. We show the existence and uniqueness (up to translations) of one-dimensional minimizers in t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.02812","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"}