Sobolev space of maps with higher dimensional target?

Question

Consider the space $W^{2,2}(\Omega)$ where $\Omega = [0,1]^3$.

This space contains continuous functions of the form $f:\Omega \to \mathbb{R}$.

I am curious, can a Sobolev space be constructed by maps of the form $ f:\Omega \to \mathbb{R}^3 $ ?

Answer 1

Of course. In general, $W^{k,p}(\Omega;\mathbb R^n)$ is the space of (equivalence classes of) functions $f\colon\Omega\to\mathbb R^n$ such that $D^\alpha f_j\in L^p(\Omega)$ for all multi-indices $\alpha$ with $|\alpha|\leq k$ and $j\in\{1,\dots,n\}$. This means that $f\colon \Omega\to\mathbb R^n$ belongs to $W^{k,p}(\Omega;\mathbb R^n)$ if and only if all the coordinate functions $f_1,\dots,f_n$ belong to the scalar-valued Sobolev space $W^{k,p}(\Omega)$.