What does the gradient of the gradient ($\nabla\nabla u$) mean? For a scalar function $u$ defined on $\mathbb{R}^n$ we have this equality: $$\nabla(|\nabla u|^{p-2})=(p-2)|\nabla u|^{p-4}\nabla u\nabla\nabla u$$ My question is : what does $\nabla\nabla u$ mean? I only know the signification of the gradient of a function, but here will it be the gradient of a vector field?

This is a somewhat awkward notation for the Hessian matrix, an $n\times n$ matrix of second derivatives. The gradient $\
abla u$ is a row vector, so $\
abla u \
abla\
abla u$ is the result of vector-matrix multiplication, also a row vector.

(One usually does not use $\
abla^2 u$ for Hessian because it'd be confused with the Laplacian. Some authors write $D^2u$ for the Hessian, and even switch from $\
abla u$ to $Du$ for consistency with that.)

More generally, if $$f=\begin{pmatrix}f_1\\\f_2\\\f_3\end{pmatrix}$$ is a vector field, then $$\
abla f=\begin{pmatrix}\
abla f_1\\\\\
abla f_2\\\\\
abla f_3\end{pmatrix} =\begin{pmatrix}\frac{\partial f_1}{\partial x_1} & \frac{\partial f_1}{\partial x_2} & \frac{\partial f_1}{\partial x_3} \\\ \frac{\partial f_2}{\partial x_1} & \frac{\partial f_2}{\partial x_2} & \frac{\partial f_2}{\partial x_3} \\\ \frac{\partial f_3}{\partial x_1} & \frac{\partial f_3}{\partial x_2} & \frac{\partial f_3}{\partial x_3} \end{pmatrix} $$