{"paper":{"title":"Gradient estimate of a Neumann eigenfunction on a compact manifold with boundary","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.SP","authors_text":"Bin Xu, Jingchen Hu, Yiqian Shi","submitted_at":"2013-06-17T22:06:25Z","abstract_excerpt":"Let $e_\\l(x)$ be a Neumann eigenfunction with respect to the positive Laplacian $\\Delta$ on a compact Riemannian manifold $M$ with boundary such that $\\Delta\\, e_\\l=\\l^2 e_\\l$ in the interior of $M$ and the normal derivative of $e_\\l$ vanishes on the boundary of $M$. Let $\\chi_\\lambda$ be the unit band spectral projection operator associated with the Neumann Laplacian and $f$ a square integrable function on $M$. We show the following gradient estimate for $\\chi_\\lambda\\,f$ as $\\lambda\\geq 1$: $\\|\\nabla\\ \\chi_\\l\\ f\\|_\\infty\\leq C\\l \\|\\chi_\\l\\f\\|_\\infty+\\l^{-1}\\|\\Delta\\ \\chi_\\l\\ f\\|_\\infty$, whe"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.4033","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"}