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Gradient Estimates on Dirichlet and Neumann Eigenfunctions
Marc Arnaudon,
Anton Thalmaier,
Feng-yu Wang 
International Mathematics Research Notices
Swansea University Author:
Feng-yu Wang
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DOI (Published version): 10.1093/imrn/rny208
Abstract
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 $$c_1(D)\ss\lambda \|\phi\|_\infty\le \|\nabla \phi\|_\infty \le c_2(D) \sqrt\lambda\|\phi\|_\infty$$ holds for...
| Published in: | International Mathematics Research Notices |
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| ISSN: | 1073-7928 1687-0247 |
| Published: |
2018
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| Online Access: |
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| URI: | https://cronfa.swan.ac.uk/Record/cronfa43217 |
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| Abstract: |
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 $$c_1(D)\ss\lambda \|\phi\|_\infty\le \|\nabla \phi\|_\infty \le c_2(D) \sqrt\lambda\|\phi\|_\infty$$ holds for any Dirichlet eigenfunction $\phi$ of $-\DD$ with eigenvalue $\lambda$. In particular, when $D$ is convex with non-negative Ricci curvature, the estimate holds for $$c_1(D)= \frac 1{de},\quad c_2(D)=\sqrt{e}\left(\frac{\sqrt{2}}{\sqrt{\pi}}+\frac{\sqrt{\pi}}{4\sqrt{2}}\right).$$ Corresponding two-sided gradient estimates for Neumann eigenfunctions are derived in the second part of the paper. |
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| College: |
Faculty of Science and Engineering |