Convergence of $f(u_n) \to f(u)$ when $u_n \to u$ in $L^p$/Sobolev spaces

Question

I got a function $f\colon \mathbb{R} \to \mathbb{R}$ which is such that $0 \leq f \leq 1$ and it is a smooth function. Suppose $u_n \to u$ in $H^1(\Omega)$. $\Omega$ is a bounded smooth domain

What is the best convergence I can get for $f(u_n) \to f(u)$?

Using the DCT I can obtain $f(u_n) \to f(u)$ in $L^p(\Omega)$ for every $p < \infty$. Is there something better like $p=\infty$ in this case?

Answer 1

If the dimension of $\Omega$ is bigger than $1$, you cannot get convergence in $L^\infty(\Omega)$ without further assumptions.

Here is a counterexample:

Let us take $v \in H^1(\Omega) \setminus L^\infty(\Omega)$ such that $v(\Omega) = [0,\infty)$ and set $u_n = v/n$. Then, $u_n \to 0$ in $H^1(\Omega)$. With $f(x) = \sin(x)$ one can check that $f(u_n)$ takes values in all of $[0,1]$. Hence, it cannot converge in $L^\infty(\Omega)$.