What does $Du$ mean in a differential equation?

Question

I'm very interested in the following work: http://maths-people.anu.edu.au/~andrews/HSU_Survey141105.pdf . Unfortunately, the author uses (in this and other papers I'm interested in) the notation $Du$. I'm unfamiliar with this notation and it isn't explicitly defined (presumably because it's an accepted convention in differential geometry or PDEs or something).

The equations tend to look something like,

$\mathcal{L} = \left[a(|Du|)\frac{u_i u_j}{|Du|^2} + \dots \right]D_i D_j u$

Does anyone know if $D$ is a gradient, covariant derivative, etc. and then $D_i$ the derivative of the $i^{th}$ coordinate or something?

Edit::

I have just discovered that further into the paper the author defines $D$ as a diameter. However, this still does not explain the indexed version and I continue to be confused as to the actual meaning of this $D$.

Answer 1

$D$ is indeed a gradient-type operator. When the term in brackets is $\delta_{ij}$, the author identifies this with the classical heat equation. This could only makes sense if $D_i D_i = \sum_i \frac{d^2}{d x_i^2}$ where the sum is implied through Einstein notation.

I think this is the answer, but if someone could back me up, I'd appreciate it.