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Chinese Annals of Mathematics, Series B

, Volume 36, Issue 6, pp 991–1000 | Cite as

The gradient estimate of a Neumann eigenfunction on a compact manifold with boundary

  • Jingchen HuEmail author
  • Yiqian Shi
  • Bin Xu
Article

Abstract

Let eλ(x) be a Neumann eigenfunction with respect to the positive Laplacian Δ on a compact Riemannian manifold M with boundary such that Δe λ = λ2 e λ in the interior of M and the normal derivative of e λ vanishes on the boundary of M. Let χλ be the unit band spectral projection operator associated with the Neumann Laplacian and f be a square integrable function on M. The authors show the following gradient estimate for χλ f as \(\lambda \geqslant 1:{\left\| {\nabla {\chi _\lambda }{\kern 1pt} \left. f \right\|} \right._\infty } \leqslant C\left( {\lambda \left\| {{\chi _\lambda }{{\left. f \right\|}_\infty } + \left. {{\lambda ^{ - 1}}} \right\|} \right.\Delta {\chi _\lambda }{{\left. f \right\|}_\infty }} \right)\), where C is a positive constant depending only on M. As a corollary, the authors obtain the gradient estimate of eλ: For every λ ≥ 1, it holds that \(\left\| {\nabla {{\left. {{e_\lambda }} \right\|}_\infty } \leqslant C\left. \lambda \right\|} \right.{\left. {{e_\lambda }} \right\|_\infty }\).

Keywords

Neumann eigenfunction Gradient estimate 

2000 MR Subject Classification

35P20 35J05 

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Copyright information

© Fudan University and Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Key Laboratory of MathematicsUSTC, Chinese Academy of SciencesHefeiChina
  2. 2.School of Mathematical SciencesUniversity of Science and Technology of ChinaHefeiChina

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