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 $|\nabla e_\l|\infty\leq C\,\l\, |e\l|_\infty$.