Let $f:\mathbb{R} \to \mathbb{R}$ be a smooth function with $|f'(x)| \leq C$ for all $x$ and $f(0)=0$.


Suppose $u\_n \to u$ in $H^{\frac 12}(\partial\Omega)$, where $\Omega$ is a bounded domain of class $C^2$. Here, I use the norm
$$|u|\_{H^{\frac 12}(\partial \Omega)}^2 = |u|^2\_{L^2(\partial\Omega)} + \int\_{\partial\Omega}\int\_{\partial\Omega} \frac{|u(x)-u(y)|^2}{|x-y|^n}.$$


Does it follow that $f(u\_n) \to f(u)$ in $H^{\frac 12}(\partial\Omega)$ (at least for a subsequence)?




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Note that $f(u) \in H^{\frac 12}$ by the Lipschitz nature of $f$. The only issue is in the convergence of the seminorm, which I can't seem to prove.



If $u_n \to u$ in $H^{\frac 12}(\partial\Omega)$, does $f(u_n) \to f(u)$ in $H^{\frac 12}(\partial\Omega)$ for $f$ Lipschitz?
**Yes.**
It might be more convenient to think of $H^{1/2}(\partial\Omega)$ as a quotient space:
$$
H^{1/2}(\partial\Omega)=H^1(\Omega)/H^1\_0(\Omega).
$$
Let $T:H^1(\Omega)\to H^{1/2}(\partial\Omega)$ be the corresponding quotient map (the trace map).
Since $u\_n\to u$ in $H^{1/2}(\partial\Omega)$, there is are functions $v\_n,v\in H^1(\Omega)$ so that $v\_n\to v$ in $H^1(\Omega)$ and $T(v\_n)=u\_n$ and $T(v)=u$.
The trace map and composing with $f$ commute with each other and $T$ is continuous, so it suffices to show that $f\circ v\_n\to f\circ v$ in $H^1(\Omega)$.
It follows from theorems 1 and 5 in [this paper](http://users.minet.uni-jena.de/~sickel/japan%28v4%29.pdf) that $H^1(\Omega)\ni w\mapsto f\circ w\in H^1(\Omega)$ is continuous, from which the convergence follows.
It suffices that $f$ is Lipschitz; smoothness is not required.


