Let $\Omega$ be a bounded open set in $\mathbb{R}^n$ and $f \in W^{1,p} (\partial \Omega)$. Can $f$ be extended to a function $u \in W^{1,p}(\Omega)$ such that $u|\_{\partial \Omega}=f$ and


$$\lVert u\rVert\_{W^{1,p}(\Omega)}\leq C\lVert f\rVert\_{W^{1,p}(\partial \Omega)}?$$


What are the minimal assumptions that guarantee such continuous extensions? 



Continuous extension of functions

> 
> **Theorem.** If $\Omega\subset\mathbb{R}^n$, $n\geq 2$, is a bounded and smooth domain, then there is a bounded extension operator $$ 
>  E:W^{1,p}(\partial\Omega)\to W^{1,q}\cap C^\infty(\Omega), \quad
>  \text{where $1<p<\infty$ and $q=\frac{np}{n-1}$.} $$
> 
> 
> 


Note that $q>p$ so $W^{1,q}(\Omega)\subset W^{1,p}(\Omega)$. Fore more details and link to a proof see: <https://mathoverflow.net/a/322635/121665>


