Estimées de gradients de fonctions propres de Dirichlet

ARNAUDON, Marc; THALMAIER, Anton; WANG, Feng-Yu

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ARNAUDON, Marc
Institut de Mathématiques de Bordeaux [IMB]

THALMAIER, Anton
Mathematics Research Unit

WANG, Feng-Yu
Tianjin University [TJU]

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Document de travail - Pré-publication

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

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AMS subject Classification: 35P20

58J65

60H30

Keywords: Eigenfunction

gradient estimate

diffusion process

curvature

second fundamental form

second fundamental

form

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