I'm trying to understand the following estimation: $ - \int_U u \left( \Delta u |Du|^{p-2} + (p-2) (\nabla u^T D^2 u \nabla u) |Du|^{p-4})\right) \leq C \int_U u |Du|^{p-2} |D^2 u| dx$
taken from Question $5.9$ - Evans PDE $2$nd edition.
I don't see how the other terms can be estimated in an appropriate way.
Maybe someone can help me to understand it?
The estimate should have $|u|$ instead of $u$. You have $|\Delta u|\le C|\nabla^2u|$ and $|\nabla u^t D^2u \nabla u|\le C |\nabla u|^2 | D^2u |$. Why does this bother you?