If i take $v\in H^1(\Omega)$ where
$$
H^1(\Omega)=\{u\in L^2(\Omega), \frac{\partial u}{\partial x\_i}\in L^2(\Omega), i=1,\ldots,N\}
$$


$\Omega$ is bounded open set from $\mathbb{R}^N$ 


What is the sufficient condition for a real function $f:\mathbb{R}\rightarrow \mathbb{R}$ to have that $f\circ v\in H^1(\Omega)$ 


Thank you 



An acting condition for a superposition operator from $H^1(\Omega)$ to $H^1(\Omega)$
The following is stated in the paper


*Moshe Marcus and Victor J. Mizel*, MR 531975 [**Complete characterization of functions which act, via superposition, on Sobolev spaces**](http://dx.doi.org/10.2307/1998689), *Trans. Amer. Math. Soc.* **251** (1979), 187--218.


as Theorem 1 (note that $T\_f$ is the superposition operator corresponding to $f$ and that the paper assumes $\Omega$ to satisfy the cone condition):



> 
> Suppose that $\Omega$ is bounded. Let $f \colon \mathbb R^m \to \mathbb R$ be a Borel function and let $p$, $r$ be two such that $1 \le r \le p < N$. Then $T\_f$ maps $W^{1,p}(\Omega)^m$ into $W^{1,r}(\Omega)$ if and only if the following conditions hold:
> 
> 
> 1. $f$ is locally Lipschitz in $\mathbb R^m$
> 2. The first order partial derivatives of $f$ satisfy the inequality
>  $$ \left| \frac {\partial f}{\partial \xi\_i}(\xi) \right| \le a\_0(1+|\xi|^\nu)\ \text{a.e. in $\mathbb R^m$, $i = 1,\dotsc,m$}$$
>  where $a\_0$ is a constant and $\nu = N(p-r)/(r(N-p))$.
> 
> 
> If $N < p$ (or $N = 1$ and $1 \le p$) and $1 \le r \le p$ then $T\_f$ maps $W^{1,p}(\Omega)^m$ into $W^{1,r}(\Omega)$ if and only if condition 1 holds.
> 
> 
> 


The theorem goes on to state that under these circumstances $T\_f$ is a continuous operator; bounds are established for its image, too. The case of unbounded $\Omega$ is considered in the same paper, see Theorem 3.


