{"paper":{"title":"Layered solutions to the vector Allen-Cahn equation in $ R^2$. Characterization of minimizers and a new approach to heteroclinic connections","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Giorgio Fusco","submitted_at":"2016-09-17T09:26:48Z","abstract_excerpt":"Let $W:R^m\\rightarrow R$ be a nonnegative potential with exactly two nondegenerate zeros $a_-\\neq a_+\\in R^m$. We assume that there are$ N\\geq 1$ distinct heteroclinic orbits connecting $a_-$ to $a_+$ represented by maps $ u_1,\\ldots,u_N$ that minimize the one-dimensional energy $J_R(u) =\\int_R(\\frac{\\vert u^\\prime\\vert^2}{2}+W(u))ds$. We first consider the problem of characterizing the minimizers $u:R^n\\rightarrow R^m$ of the energy $\\mathcal{J}_\\Omega(u) =\\int_\\Omega(\\frac{\\vert\\nabla u\\vert^2}{2}+W(u))dx$. Under a nondegeneracy condition on $ u_1,\\ldots,u_N $ and in two space dimensions, we"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.05306","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"}