Let $f_n\colon \mathbb{R} \to \mathbb{R}$ be such that $f_n \to f$ uniformly on compact subsets of $\mathbb{R}$. Furthermore $f(0)=f_n(0)=0$. These functions are smooth and bijective. Also $f_n'$ is bounded away from $0$ and $\infty$ by constants (that depend on $n$).
Let $u \in L^\infty(\Omega)$, where $\Omega$ is a bounded domain. Does it follow that
- $$f_n(u) \to f(u)\quad\text{in $L^2(\Omega)$}?$$
- $$|f_n(u)| \leq C$$ where $C$ is a constant that does not depend on $n$?
1 would imply 2 but I don't see how to use DCT there.
Most of those hypotheses aren't needed. Suppose $u:\Omega \to \mathbb {R}$ is bounded and measurable. Then $u(\Omega )$ is contained in a compact set $K.$ Suppose $f_n$ is a sequence of bounded continuous functions on $K$ that converges uniformly to $f$ on $K.$ Then $f$ is continuous on $K.$ Because these functions are continuous, the functions $f_n\circ u, f\circ u$ are measurable on $\Omega.$ We have $f_n\circ u\to f\circ u$ uniformly on $\Omega,$ hence in any $L^p(\Omega)$ because $\Omega$ is bounded. That proves 1. Any uniformly convergent sequence of bounded functions is uniformly bounded. That gives 2.