The gradient estimate of a Neumann eigenfunction on a compact manifold with boundary
The gradient estimate of a Neumann eigenfunction on a compact manifold with boundary
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DOI：
10.1007/s11401-015-0924-6
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发表时间：
2013-06
期刊：
影响因子：
--
通讯作者：
Jingchen Hu;Yiqian Shi;Bin Xu
中科院分区：
文献类型：
--
作者：
Jingchen Hu;Yiqian Shi;Bin Xu
Leteλ(x) be a Neumann eigenfunction with respect to the positive Laplacian Δ on a compact Riemannian manifoldMwith boundary such that Δeλ= λ2eλin the interior ofMand the normal derivative ofeλvanishes on the boundary ofM. Let χλ be the unit band spectral projection operator associated with the Neumann Laplacian andfbe a square integrable function onM. The authors show the following gradient estimate for χλfas $$\lambda \geqslant 1:{\left\| {\nabla {\chi _\lambda }{\kern 1pt} \left. f \right\|} \right._\infty } \leqslant C\left( {\lambda \left\| {{\chi _\lambda }{{\left. f \right\|}_\infty } + \left. {{\lambda ^{ - 1}}} \right\|} \right.\Delta {\chi _\lambda }{{\left. f \right\|}_\infty }} \right)$$, whereCis a positive constant depending only onM. As a corollary, the authors obtain the gradient estimate ofeλ: For every λ ≥ 1, it holds that.
Leteλ(x) be a Neumann eigenfunction with respect to the positive Laplacian Δ on a compact Riemannian manifoldMwith boundary such that Δeλ= λ2eλin the interior ofMand the normal derivative ofeλvanishes on the boundary ofM. Let χλ be the unit band spectral projection operator associated with the Neumann Laplacian andfbe a square integrable function onM. The authors show the following gradient estimate for χλfas $$\lambda \geqslant 1:{\left\| {\nabla {\chi _\lambda }{\kern 1pt} \left. f \right\|} \right._\infty } \leqslant C\left( {\lambda \left\| {{\chi _\lambda }{{\left. f \right\|}_\infty } + \left. {{\lambda ^{ - 1}}} \right\|} \right.\Delta {\chi _\lambda }{{\left. f \right\|}_\infty }} \right)$$, whereCis a positive constant depending only onM. As a corollary, the authors obtain the gradient estimate ofeλ: For every λ ≥ 1, it holds that.