Given $u\in W^{1,2}(\mathbb{R}^n)$ and a smooth bump function $\eta\in C^\infty(\mathbb{R}^n)$ with $\eta(x)=0$ for $||x||\ge 1$, the mollified functions $u_\epsilon \in C^\infty(\mathbb{R}^n)$ for each $\epsilon>0$ given by $u_\epsilon(x):=\int_{\mathbb{R}^n} u(y) \eta\left(\frac{||y-x||}{\epsilon}\right)dy$ satisfies $\lim_{\epsilon\to 0} ||u_\epsilon-u||_{L^2(\mathbb{R}^n)} = 0$ and $$||\,|\nabla u_\epsilon|\,||_{L^2(\mathbb{R}^n)} \le ||\,|\nabla u|\,||_{L^2(\mathbb{R}^n)}.$$
I want to construct another function $\hat{u}_\epsilon$ which is a "local mollification" of $u$. That is, given open sets $\Omega_1,\Omega_2\subset \mathbb{R}^n$ with $\overline{\Omega_1} \cap \overline{\Omega_2}=\emptyset$, I want to construct , with the following properties:
- $\hat{u}_\epsilon$ is smooth inside $\Omega_1$,
- $\hat{u}_\epsilon=u$ inside $\Omega_2$,
- $\lim_{\epsilon\to 0} ||u_\epsilon-u||_{L^2(\mathbb{R}^n)} = 0$,
- $||\,|\nabla \hat{u}_\epsilon|\,||_{L^2(\mathbb{R}^n)} \le ||\,|\nabla u|\,||_{L^2(\mathbb{R}^n)}$.
Is this possible for all such $u$? Is there an explicit construction for $\hat{u}_\epsilon$?