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| The Gradient Estimate of a Neumann Eigenfunction on a CompactManifold with Boundary |
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Citation： |
Jingchen HU,Yiqian SHI,Bin XU.The Gradient Estimate of a Neumann Eigenfunction on a CompactManifold with Boundary[J].Chinese Annals of Mathematics B,2015,36(6):991~1000 |
| Page view： 1166
Net amount： 1079 |
Authors： |
Jingchen HU; Yiqian SHI;Bin XU |
Foundation： |
This work was supported by the National Natural Science
Foundation of China (Nos.\,10971104, 11271343, 11101387), the Anhui
Provincial Natural Science Foundation (No.\,1208085MA01) and the
Fundamental Research Funds for the Central Universities
(Nos.\,WK0010000020, WK0010000023, WK3470000003). |
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| Abstract： |
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 $f$ be a square
integrable function on $M$. The authors show the following gradient
estimate for $\chi_\lambda\,f$ as $\lambda\geq 1$: $\|\nabla
\chi_\l\ f\|_\infty\leq C(\l \|\chi_\l\ f\|_\infty+\l^{-1}\|\Delta
\chi_\l\ f\|_\infty)$, where $C$ is a positive constant depending
only on $M$. As a corollary, the authors obtain the gradient
estimate of $e_\l$: For every $\l\geq 1$, it holds that $\|\nabla
e_\l\|_\infty\leq C\,\l\, \|e_\l\|_\infty$. |
Keywords： |
Neumann eigenfunction, Gradient estimate |
Classification： |
35P20, 35J05 |
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