{"paper":{"title":"Local Minimizers of the Ginzburg-Landau Functional with Prescribed Degrees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Micka\\\"el Dos Santos (ICJ)","submitted_at":"2011-11-07T13:31:41Z","abstract_excerpt":"We consider, in a smooth bounded multiply connected domain $\\dom\\subset\\R^2$, the Ginzburg-Landau energy $\\d E_\\v(u)=1/2\\int_\\dom{|\\n u|^2}+\\frac{1}{4\\v^2}\\int_\\dom{(1-|u|^2)^2}$ subject to prescribed degree conditions on each component of $\\p\\dom$. In general, minimal energy maps do not exist \\cite{BeMi1}. When $\\dom$ has a single hole, Berlyand and Rybalko \\cite{BeRy1} proved that for small $\\v$ local minimizers do exist. We extend the result in \\cite{BeRy1}: $\\d E_\\v(u)$ has, in domains $\\dom$ with $2,3,...$ holes and for small $\\v$, local minimizers. Our approach is very similar to the one"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.1571","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"}