Knowing that $\omega\Subset\Omega\subset\mathbb{R}^2$ (compactly included) are two open and bounded sets with $C^2$ boundary, is it true that for any function $\phi_0:\overline{\omega}\to\mathbb{R},\ \phi_0\in C^1(\overline{\omega})$ ($\overline{\omega}$ is the closure of $\omega$) we can find an extension $\phi:\Omega\to\mathbb{R}$ with $\phi\in C^1_{c}(\Omega)$ (compactly supported in $\Omega$)?

**Motivation**

If this type of result is true than we can obtain simple formulas for perimeter of implicitly defined curves in $\mathbb{R}^2$, putting $\phi_0$ the unit outer normal vector to a regular curve (which is defined in a neighborhood of the boundary of $\omega$). See here: Perimeter continuity of $BV$ sets on any sequence from $W^{1,1}$

**What did I do?**

I proved by standard methods (using cut-off functions and convolution) that we can obtain a mollifying sequence $\phi_n,\ n\in\mathbb{N}^*$ compactly supported in $\Omega$ that tends to $\phi_0$ in $L^1(\Omega)$, but I cannot prove that we can indeed have an extension.