{"paper":{"title":"Local minimality of $\\mathbb{R}^N$-valued and $\\mathbb{S}^N$-valued Ginzburg-Landau vortex solutions in the unit ball $B^N$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Luc Nguyen, Radu Ignat","submitted_at":"2021-11-15T10:52:37Z","abstract_excerpt":"We study the existence, uniqueness and minimality of critical points of the form $m_{\\varepsilon,\\eta}(x) = (f_{\\varepsilon,\\eta}(|x|)\\frac{x}{|x|}, g_{\\varepsilon,\\eta}(|x|))$ of the functional \\[ E_{\\varepsilon,\\eta}[m] = \\int_{B^N} \\Big[\\frac{1}{2} |\\nabla m|^2 + \\frac{1}{2\\varepsilon^2} (1 - |m|^2)^2 + \\frac{1}{2\\eta^2} m_{N+1}^2\\Big]\\,dx \\] for $m=(m_1, \\dots, m_N, m_{N+1}) \\in H^1(B^N,\\mathbb{R}^{N+1})$ with $m(x) = (x,0)$ on $\\partial B^N$. We establish a necessary and sufficient condition on the dimension $N$ and the parameters $\\varepsilon$ and $\\eta$ for the existence of an escaping "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2111.07669","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2111.07669/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}