Sobolev spaces for vector-valued functions

Question

Let $ \Omega \subseteq \mathbb{R}^3$ be open. How is defined the space $W^{1,p}(\Omega)$ for vector valued functions $f:\Omega \to \mathbb{R}^3 $? What is the norm in $W^{1,p}(\Omega)$?

Answer 1

Note that the Sobolev $p$-norm for scalar-valued functions is \begin{align*} \|f\|_{W^{1,p}}^p=\int_\Omega |f|^p +|\nabla f |^p dm. \end{align*} Here, you should not hesitate to think of $|\cdot|$ as the underlying Euclidean norm on $\mathbb{R}$ measuring the size of the output of the function and its gradient. Replacing this with any norm $\|\cdot\|_E$ (recall they are all equivalent) on $\mathbb{R}^n$ gives you the Sobolev norm on vector valued functions as \begin{align*} \|f\|_{W^{1,p}}^p=\int_\Omega \|f\|_E^p +\|\nabla f \|_E^p dm. \end{align*} In particular, I guess the standard choice would be to to choose either \begin{align*} \|x\|_E =\sqrt{\sum_{i=1}^n |x_i|^2}, \end{align*} which is MaoWao's suggestion, or \begin{align*} \|x\|_E =\sum_{i=1}^n |x_i|, \end{align*} which gives the Sobolev norm mentioned in the comment by daw.