{"paper":{"title":"Multiplicity of layered solutions for Allen-Cahn systems with symmetric double well potential","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Francesca G. Alessio, Piero Montecchiari","submitted_at":"2013-09-12T10:40:24Z","abstract_excerpt":"We study the existence of solutions $u:\\R^{3}\\to\\R^{2}$ for the semilinear elliptic systems \\begin{equation}\\label{eq:abs} -\\Delta u(x,y,z)+\\nabla W(u(x,y,z))=0, \\end{equation} where $W:\\R^{2}\\to\\R$ is a double well symmetric potential. We use variational methods to show, under generic non degenerate properties of the set of one dimensional heteroclinic connections between the two minima $\\a_{\\pm}$ of $W$, that (\\ref{eq:abs}) has infinitely many geometrically distinct solutions $u\\in C^{2}(\\R^{3},\\R^{2})$ which satisfy $u(x,y,z)\\to \\a_{\\pm}$ as ${x\\to\\pm\\infty}$ uniformly with respect to $(y,z"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.3104","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"}