{"paper":{"title":"Catenoidal layers for the Allen-Cahn equation in bounded domains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.AP","authors_text":"Juncheng Wei, Manuel del Pino, O. Agudelo","submitted_at":"2015-04-21T04:36:52Z","abstract_excerpt":"In this paper we present a new family of solutions to the singularly perturbed Allen-Cahn equation $\\alpha^2 \\Delta u + u(1-u^2)=0, \\quad \\hbox{in }\\Omega\\subset \\R^N $\nwhere $N=3$, $\\Omega$ is a smooth bounded domain and $\\A>0$ is a small parameter. We provide asymptotic behavior which shows that, as $\\alpha \\to 0$, the level sets of the solutions collapse onto a bounded portion of a complete embedded minimal surface with finite total curvature that intersects orthogonally $\\partial \\Omega$ of the domain and that is non-degenerate respect to $\\Omega$. We provide explicit examples of surfaces "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.05301","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"}