Gradient Estimates on Dirichlet Eigenfunctions

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 $c_1(D)\sqrt{\lambda}\|\phi\|_\infty \le \|\nabla \phi\|_\infty\le c_2(D)\sqrt{\lambda} \|\phi\|_\infty$ 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.
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  • HAL Id : hal-01625890, version 3
  • ARXIV : 1710.10832

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Marc Arnaudon, Anton Thalmaier, Feng-Yu Wang. Gradient Estimates on Dirichlet Eigenfunctions. 2018. ⟨hal-01625890v3⟩

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