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Gradient Estimates on Dirichlet and Neumann Eigenfunctions

Marc Arnaudon, Anton Thalmaier, Feng-yu Wang Orcid Logo

International Mathematics Research Notices

Swansea University Author: Feng-yu Wang Orcid Logo

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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...

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Published in: International Mathematics Research Notices
ISSN: 1073-7928 1687-0247
Published: 2018
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URI: https://cronfa.swan.ac.uk/Record/cronfa43217
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first_indexed 2018-08-04T03:57:49Z
last_indexed 2018-11-26T20:15:28Z
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spelling 2018-11-26T14:31:06.2302382 v2 43217 2018-08-04 Gradient Estimates on Dirichlet and Neumann Eigenfunctions 6734caa6d9a388bd3bd8eb0a1131d0de 0000-0003-0950-1672 Feng-yu Wang Feng-yu Wang true false 2018-08-04 SMA 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. Journal Article International Mathematics Research Notices 1073-7928 1687-0247 4 9 2018 2018-09-04 10.1093/imrn/rny208 COLLEGE NANME Mathematics COLLEGE CODE SMA Swansea University 2018-11-26T14:31:06.2302382 2018-08-04T01:19:21.0100009 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics Marc Arnaudon 1 Anton Thalmaier 2 Feng-yu Wang 0000-0003-0950-1672 3 0043217-04082018012052.pdf 18ATW.pdf 2018-08-04T01:20:52.0070000 Output 293406 application/pdf Accepted Manuscript true 2019-09-04T00:00:00.0000000 true eng
title Gradient Estimates on Dirichlet and Neumann Eigenfunctions
spellingShingle Gradient Estimates on Dirichlet and Neumann Eigenfunctions
Feng-yu Wang
title_short Gradient Estimates on Dirichlet and Neumann Eigenfunctions
title_full Gradient Estimates on Dirichlet and Neumann Eigenfunctions
title_fullStr Gradient Estimates on Dirichlet and Neumann Eigenfunctions
title_full_unstemmed Gradient Estimates on Dirichlet and Neumann Eigenfunctions
title_sort Gradient Estimates on Dirichlet and Neumann Eigenfunctions
author_id_str_mv 6734caa6d9a388bd3bd8eb0a1131d0de
author_id_fullname_str_mv 6734caa6d9a388bd3bd8eb0a1131d0de_***_Feng-yu Wang
author Feng-yu Wang
author2 Marc Arnaudon
Anton Thalmaier
Feng-yu Wang
format Journal article
container_title International Mathematics Research Notices
publishDate 2018
institution Swansea University
issn 1073-7928
1687-0247
doi_str_mv 10.1093/imrn/rny208
college_str Faculty of Science and Engineering
hierarchytype
hierarchy_top_id facultyofscienceandengineering
hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
hierarchy_parent_title Faculty of Science and Engineering
department_str School of Mathematics and Computer Science - Mathematics{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Mathematics
document_store_str 1
active_str 0
description 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.
published_date 2018-09-04T03:54:29Z
_version_ 1763752719530590208
score 11.035634

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