$C^{\infty}$ reconciliation of two functions

Question

Let $\Omega$ be an open subset of $\mathbb{R}^d$, $K$ a compact subset of $\Omega$ and $\phi \in C^\infty(\Omega)$. Given $\epsilon > 0$ and denoting $K_\epsilon = \{ x \in K : d(x,K^c) \geq \epsilon \}$, is it possible to find $\psi \in C^\infty(\Omega)$ such that $\psi = \phi$ on $K_\epsilon$ and $\psi = 0$ on $K^c$? I feel like there is some space between $K_\epsilon$ and $K^c$ so this is reasonable but I cannot see why it should be true.

Answer 1

Let $\rho\in C^\infty_c(\Omega)$ such that $\int \rho(x) \, dx=1$ and $\operatorname{supp}\rho\subset B_{\epsilon/2}(0).$ Then let $\rho_{\epsilon}=\rho*\mathbf{1}_{K_{\epsilon/2}},$ where $\mathbf{1}_A$ is the indicator function on the set $A.$ Now $\rho_\epsilon\in C^\infty_c(\Omega)$ equals $1$ on $K_{\epsilon}$ and equals $0$ on $K^c.$ Therefore $\phi\rho_\epsilon$ is the kind of function that you are looking for.