$u\in W^{1,1}(\Omega)$, $f\in C^1(\mathbb{R})$, but: $f\circ u\notin W^{1,1}(\Omega)$

Question

I'm searching for an example for $u\in W^{1,1}(\Omega)$, $f\in C^1(\mathbb{R})$ such that the composition $f\circ u\notin W^{1,1}(\Omega)$, where $\Omega\subset \mathbb{R}^n$ is open, bounded.

I know that if $f\in C^1(\mathbb{R})\cap W^{1,\infty}(\mathbb{R})$, then it's not possible to find such an example. And I know that $\frac{1}{x}$ is not in $W^{1,1}(0,1)$ for example, so that I tried to contruct something similar (for instance I tried to find $f$ and $u$ as above such that $(f\circ u)(x)=\frac{1}{x}$) ...

However, I wasn't successful in finding such an example. I appreciate hints/ideas for suitable functions.

Answer 1

Consider $\Omega = \biguplus_n \Omega_n $, where $\Omega_n = (2^{-(n+1)}, 2^{-n}) $ and define $u : \Omega \to \Bbb {R} $ by $u \equiv 2^n / n^2$ on $\Omega_n $.

Finally choose $f (x) = x^2$. I will leave the verification of the details to you.