To prove the Rellich-Kondrachov Theorem it is used the following statement
If $u\in W^{1,1}(\Omega)$, with $\Omega \subset \mathbb{R}^N$ open, bounded and s.t. $\partial \Omega$ is $C^1$, then $||\rho_\epsilon * u - u||_{L^1(\Omega)} \le \epsilon ||\nabla u|| _{L^1(\Omega,\mathbb{R}^N)}$ for all $\epsilon >0$, where $\rho_\epsilon$ is the Friedrichs’ mollifier.
I should find an example to prove that a similar statement is false if I replace the hypothesis $u\in W^{1,1}(\Omega)$ with $u\in L^1(\Omega)$. Precisely, I should prove that the following statement is false
If $u\in L^1(\Omega)$, with $\Omega \subset \mathbb{R}^N$ open, bounded and s.t. $\partial \Omega$ is $C^1$ then it exists a constant $C>0 $ s.t. $||\rho_\epsilon * u - u||_{L^1(\Omega)} \le C\epsilon$ for all $\epsilon >0$.
My attempt was to take a function assuming just the two values $1$ and $-1$ oscillating a lot, but I’m having some trouble to do it. Can someone please help me, or indicating me a reference?
