I am thinking about the recently noticed p-Laplace system
$div\left(|Du|^{p-2}D u_i \right)= 0$ in $E\subset \mathbb{R}^n$, $1\leq i \leq n $
where $u:E\subset \mathbb{R}^m \to \mathbb{R}^n$. What is $Du$ exactly? It should be a matrix instead of a vector (gradient), right? If so, which norm is meant with $|Du|$? I would assume that this is the usual spectral norm used on matrix spaces but I am not certain. I understood the scalar case, i.e. $n=1$, but I have never before seen any system like this.
Thanks for the help!