Working with norms I was hoping to get some help with being able to properly work with norms and derivatives so I can actually understand my PDE course. We are currently working on Sobolev spaces. 
Example, I want to show that:
$$-\int_{U} u \Delta u dx \leq C \int_{U}|u||D^2u|dx$$
$u \in C_{c}^{\infty}(U)$ with $U$ bounded.
I get to:
$$-\int_{U} u \Delta u dx \leq \int_{U} |u||\Delta u|dx$$
I know it is not very far at all, but I am so confused. I am missing some key skills in multivariable calculus.
My main question: What is $|D^2u|$? I thought that $D^2u$ was the hessian, and I'm confused about taking the norm.
If you have any links that would help me better understand operations with norms and $D^ku$ I would greatly appreciate it.
I'm really trying, but its just not clicking. Its really frustrating to be undone by the simpler concepts in an extremely theoretical PDE course.

It looks like you're using a book using notation similar to Evans. If you are, check Appendix A, in the section Notation for derivatives, where we see
$$
|D^k u| = \left(\sum_{|\alpha|=k} |D^\alpha u|^2\right)^{1/2}.
$$
With this it should be pretty clear why $|\Delta u| \leq C|D^2 u|$.
Notation for derivatives is a bit weird across books. When in doubt, check your appendix or symbol glossary.
