we defined the norm in a Sobolev space $W^{k,p}$ as $||u||_{W^{k,p}(\Omega)}:=\sum_{|\alpha|\leq k}||D^{\alpha}u||_{L^p{(\Omega)}}$ but we also said that with $W^{2,2}(\Omega)=H^2(\Omega)$ it is $||u||_{H^2}=||u||_{L^2}+||\nabla u||_{L^2}+||\Delta u||_{L^2}$.
Question: when do you use $D^1u$ and when $\nabla u$ (or $D^2u$, $\Delta u$)? Especially in the Poincare-inequality for example. It was mentioned that for smooth functions with compact support the identity $||D^2u||_{L^2}^2=||\Delta u||_{L^2}^2$ holds. Why? How to prove it?
Thanks!