Suppose $0<u \in C^1(\overline{\Omega})$, $\Omega \subset \mathbb{R}^N$ bounded regular domain. If the normal derivative of $u$ satisfies
$\max_{\partial\Omega}\dfrac{\partial u}{\partial \eta} < 0$ (here $\eta$ is the outward normal vector)
is it true that $\sup_{V_\delta}\dfrac{\partial u}{\partial \eta} < 0$, where $V_\delta = \{x \in \Omega \hspace{0.1cm}; \hspace{0.1cm} dist(x,\partial\Omega) < \delta\}$, for some $\delta >0$ small enough?