Sobolev spaces and weak derivatives. Meaning of $D^2$

Question

Im getting confused on the notation. Let $u \in W^{2,2}(\Omega)$, $\Omega$ open and bounded of $\mathbb{R}^n$. When we write $D^2u$ is this equivalent to $(D^{v_i,v_j})_{i,j}$, where $v_i = (0,... , 1, ..., 0)$ the 1 going in pos $i$. So we would have $D^2u: \Omega \rightarrow \mathbb{R}^n \times \mathbb{R}^n$ or is it some kind of weak-derivative equivalent of the laplace operator?

Answer 1

$D^2 u $ is the notation for the Hessian matrix of $u$ (https://en.wikipedia.org/wiki/Hessian_matrix)