# Extension of outer unit normal vector to interior

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 $$C/\delta;C/\log\delta..$$

• @PiotrHajlasz, Thanks, I will think about it more.
– H-H
May 22, 2020 at 4:00
• I think this question would be more appropriate for the Mathematics Stack Exchange as it is a standard exercise. May 22, 2020 at 12:55

If $$\partial\Omega\in C^2$$, then the answer is yes. We need $$C^2$$ as this condition implies that the vector normal to the boundary is $$C^1$$ (since the normal vector id defined through derivatives).
According to the collar neighborhood theorem, there is $$d>0$$ and a diffeomorphism of class $$C^1$$: $$\Phi:\partial\Omega\times(-1,1)\to (\partial\Omega)_d=\{x:\operatorname{dist}(x,\partial\Omega) Let $$\pi_1:\partial\Omega\times(-1,1)\to\partial\Omega,\ \ \pi_1(x,t)=x,$$ and $$\pi_2:\partial\Omega\times(-1,1)\to\partial\Omega,\ \ \pi_2(x,t)=t,$$ be projections on components. Then $$\tilde{X}=\eta\circ\pi_1\circ\Phi^{-1}$$ is a $$C^1$$ extension of the vector field $$\eta$$ to a neighborhood of the boundary. Let $$\phi\in C_0^\infty(-1,1), \quad \text{with} \quad \text{\phi(t)=1 for t\in [-1/2,2]} \quad \text{and \phi_\epsilon(t)=\phi(t/\epsilon).}$$ Then $$|(\phi_\epsilon)'(t)|\leq C/\epsilon$$ and hence $$X=\tilde{X}\cdot (\phi_\epsilon\circ\pi_2\circ\Phi^{-1})$$ satisfies the required condition (with $$\epsilon$$ in place of $$\delta$$).
• could you recommend a reference for just an extension $X\in C^2(\overline{\Omega}): \, |X(x)|\le 1$ in $\Omega$? Thanks in advance. Jun 17, 2021 at 10:31