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

Abstract

Let $e_\l(x)$ be a Neumann eigenfunction with respect to the positive Laplacian $\Delta$ on a compact Riemannian manifold $M$ with boundary such that $\Delta\, e_\l=\l^2 e_\l$ in the interior of $M$ and the normal derivative of $e_\l$ 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 $\lambda\geq 1$: $\|\nabla\ \chi_\l\ f\|_\infty\leq C\l \|\chi_\l\f\|_\infty+\l^{-1}\|\Delta\ \chi_\l\ f\|_\infty$, where $C$ is a positive constant depending only on $M$. As a corollary, we obtain the gradient estimate of $e_\l$: for every $\l\geq 1$, there holds $\|\nabla e_\l\|_\infty\leq C\,\l\, \|e_\l\|_\infty$.