I don't understand definition of Dirichlet energy

$$ E[u]:=\frac{1}{2} \int_\Omega \Vert \nabla u \Vert^2 $$ for $u:\Omega \to \mathbb R^n$ with $\Omega \subset \mathbb R^n$.
Let's consider for example $u: \Omega \to \mathbb R^2$. What would

$$ \Vert \nabla u \Vert^2 $$ mean? Can we say it's determinant of Jacobi Matrix of $u$ or rather quadratic sum of all components of it?

Edit: for the people wondering Dirichlet energy in general is defined as:

$$E[u]:=\frac{1}{2} \int_\Omega trace(du^t\cdot du)$$

  • 2
    $\begingroup$ $$||\nabla u||^2=\nabla u\cdot\nabla u$$ $\endgroup$
    – Mark Viola
    Feb 18, 2019 at 19:54
  • $\begingroup$ @MarkViola I'm only familiar with $\nabla u$ as gradient of function $u$ from $\mathbb R^n$ to $\mathbb R $, but according to definition above $u$ can be vector valued, in that case $\nabla u$ does not make sense to me. Is it some sort of generalisation? Thanks! $\endgroup$ Feb 18, 2019 at 20:26
  • $\begingroup$ It might be a typo in the Spanish Wikipedia page you linked, since in the English page it has $u:\Omega \to \mathbb{R}$. $\endgroup$
    – angryavian
    Feb 18, 2019 at 20:37
  • $\begingroup$ $\nabla u$ is a vector. $\endgroup$
    – Mark Viola
    Feb 18, 2019 at 20:41
  • $\begingroup$ @angryavian, I thought so aswell, but in other notes it is defined is similiar way, see Def 2.4 page 5 $\endgroup$ Feb 18, 2019 at 20:43


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