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

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    $\begingroup$ @PiotrHajlasz, Thanks, I will think about it more. $\endgroup$
    – H-H
    May 22 '20 at 4:00
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    $\begingroup$ I think this question would be more appropriate for the Mathematics Stack Exchange as it is a standard exercise. $\endgroup$ May 22 '20 at 12:55
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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 $C^1$: $$ \Phi:\partial\Omega\times(-1,1)\to (\partial\Omega)_d=\{x:\operatorname{dist}(x,\partial\Omega)<d\} $$ 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 $\delta$).

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  • $\begingroup$ could you recommend a reference for just an extension $X\in C^2(\overline{\Omega}): \, |X(x)|\le 1$ in $\Omega$? Thanks in advance. $\endgroup$
    – S. Euler
    Jun 17 at 10:31

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