{"paper":{"title":"Asymptotics for the Ginzburg-Landau equation on manifolds with boundary under homogeneous Neumann condition","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.AP","authors_text":"Da Rong Cheng","submitted_at":"2018-01-11T20:58:33Z","abstract_excerpt":"On a compact manifold $M^{n}$ ($n\\geq 3$) with boundary, we study the asymptotic behavior as $\\epsilon$ tends to zero of solutions $u_{\\epsilon}: M \\to \\mathbb{C}$ to the equation $\\Delta u_{\\epsilon} + \\epsilon^{-2}(1 - |u_{\\epsilon}|^{2})u_{\\epsilon} = 0$ with the boundary condition $\\partial_{\\nu}u_{\\epsilon} = 0$ on $\\partial M$. Assuming an energy upper bound on the solutions and a convexity condition on $\\partial M$, we show that along a subsequence, the energy of $\\{u_{\\epsilon}\\}$ breaks into two parts: one captured by a harmonic $1$-form $\\psi$ on $M$, and the other concentrating on t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.03987","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"}