Gradient Estimates on Dirichlet and Neumann Eigenfunctions

Author:

Arnaudon Marc1,Thalmaier Anton2,Wang Feng-Yu34

Affiliation:

1. Institut de Mathématiques de Bordeaux, Université de Bordeaux, Cours de la Libération, Talence Cedex, France

2. Mathematics Research Unit, University of Luxembourg, Maison du Nombre, Esch-sur-Alzette, Luxembourg

3. Center for Applied Mathematics, Tianjin University, Tianjin, China

4. Department of Mathematics, Swansea University, Singleton Park, United Kingdom

Abstract

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)\sqrt \lambda \|\phi \|_\infty \leqslant \|\nabla \phi \|_\infty \leqslant c_2(D)\sqrt \lambda \|\phi \|_\infty $ holds for any Dirichlet eigenfunction $\phi $ of $-\Delta $ with eigenvalue $\lambda $. In particular, when $D$ is convex with nonnegative Ricci curvature, the estimate holds for $c_1(D)= 1/{d\mathrm{e}}$ and $c_2(D)=\sqrt{\mathrm{e}}\left (\frac{\sqrt{2}}{\sqrt{\pi }}+\frac{\sqrt{\pi }}{4\sqrt{2}}\right ).$ Corresponding two-sided gradient estimates for Neumann eigenfunctions are derived in the second part of the paper.

Funder

Fonds National de la Recherche Luxembourg

University of Luxembourg

Publisher

Oxford University Press (OUP)

Subject

General Mathematics

Reference14 articles.

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