Gradient estimate of a Neumann eigenfunction on a compact manifold with boundary

Abstract

Let $e_{ł}(x)$ be a Neumann eigenfunction with respect to the positive Laplacian $\Delta$ on a compact Riemannian manifold $M$ with boundary such that $\Delta e_{ł}={ł}^2{e}_{ł}$ in the interior of $M$ and the normal derivative of $e_{ł}$ vanishes on the boundary of $M$. Let ${\chi }_{\lambda }$ be the unit band spectral projection operator associated with the Neumann Laplacian and $f$ a square integrable function on $M$. We show the following gradient estimate for ${\chi }_{\lambda }f$ as $\lambda \ge 1$: $\|\nabla\ χ_ł f\|_\infty\leq Cł\|χ_ł\f\|_\infty+ł^{-1}\|Δ χ_ł f\|_\infty$, where $C$ is a positive constant depending only on $M$. As a corollary, we obtain the gradient estimate of $e_{ł}$: for every $ł\ge 1$, there holds $\Vert \nabla e_{ł}{\Vert }_{\infty }\le Cł\Vert e_{ł}{\Vert }_{\infty }$.

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