I am currently reading a paper on monge-ampere equations, and in one part the author does as follows.
Let $\Omega,\Omega^*$ be two uniformly convex subsets of $\Bbb R^n$, and let $h\in C^{2,1}(\Bbb R^n)$ be a uniformly concave defining function for $\Omega^*$, i.e
$\Omega^*=\{p\in\Bbb R^n|h(p)\gt 0\}$ and $\|D h\|\ne 0$ ($D h=\nabla h$).
We may reformulate $Du(\Omega)=\Omega^*$ as $h(Du)=0$ on $\partial\Omega$.
Let $H=h(Du)$, since $H\gt 0$ in $\Omega$ and $H=0$ on $\partial\Omega$.
It is clear that for any tangent vector field $\tau$, $D_\tau H=0$, and for an inner normal vector field $\nu$, $D_\nu H\ge 0$.
I understand every thing up to this part, until the notation completely throws me, the author proceeds as follows:
$D_\tau H=h_{p_k}D_{k\tau}u=0$ on $\partial\Omega$.
$D_\nu H=h_{p_k}D_{k\nu}u\ge 0$ on $\partial\Omega$.
Any help in deciphering this notation would be greatly appreciated, since I cannot tell for instance if $h_{p_k}$ denotes a derivative or just an index.
Thank you all!
Note that in the paper, the notation implies that $p=Du$