{"paper":{"title":"Gradient Estimates on Dirichlet Eigenfunctions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Anton Thalmaier, Feng-Yu Wang (TJU), Marc Arnaudon (IMB)","submitted_at":"2017-10-30T09:49:17Z","abstract_excerpt":"By methods of stochastic analysis on Riemannian manifolds, we derive explicit constants $c\\_1(D)$ and $c\\_2(D)$ for a $d$-dimensional compact Riemannian manifold $D$ with boundary such that $c\\_1(D)\\sqrt{\\lambda}\\|\\phi\\|\\_\\infty \\le \\|\\nabla \\phi\\|\\_\\infty\\le c\\_2(D)\\sqrt{\\lambda} \\|\\phi\\|\\_\\infty$ holds for any Dirichlet eigenfunction $\\phi$ of $-\\Delta$ with eigenvalue $\\lambda$. In particular, when $D$ is convex with nonnegative Ricci curvature, this estimate holds for $c\\_1(D)=\\frac{1}{de}$ and $c\\_2(D)=\\sqrt{e}\\left(\\frac{\\sqrt{2}}{\\sqrt{\\pi}}+\\frac{\\sqrt{\\pi}}{4\\sqrt{2}}\\right)$.  Corres"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.10832","kind":"arxiv","version":3},"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"}