Suppose we have a bounded smooth domain $\Omega$ in $\mathbb{R}^n$, so there exists an outer unit normal vector field $\eta$ everywhere on the boundary. Can we extend it to the interior satisfying some conditions? To be more specific, for example, can we construct a vector field $ X$ on $\Omega$ satisfying

(1) $X|_{\partial \Omega}=\eta;$

(2) $X$ is compactly supported in $\delta$-neighborhood of boundary $\Omega_\delta=\{x\in\Omega: d(x,\partial\Omega)\leq \delta\}$ for very small $\delta$.

(3) For example,the vector field $X$ is bounded in C^1: $\|X\|_{C^1(\bar{\Omega})}\leq C/\delta$ where C is a universal constant.

As it is pointed out, I should let the boundary be at least $C^2$ such that (3) makes sense. So for simplicity, let's just consider domain with smooth boundary. My thought was to think $\eta=(\eta_1,\cdots,\eta_n)$, so $\eta_i$ are smooth functions on the boundary, we can use the partition of unity to extend these functions to functions $X_i$ in some neighborhood of boundary. Then certainly we will get bounds in $C^k$ for $X_i$. I am just wondering if we can extend it to $\delta$-neighborhood and make those bounds (such as $C^1$ norm) depending on $\delta$, for example, bounds like $C\delta; C/\delta; C\delta^2;C\log\delta...$ where $C$ is a universal constant. How do we make this work exactly?

Edit: Maybe those bounds such as $C\delta,C\delta^2,..$ don't look reasonable since we expect that the derivative should become greater when $\delta$ is smaller. So it only makes sense to expect bounds like $C/\delta;C/\log\delta..$