Let $Q$ be a linear constant coefficient second order differential operator (for example $curl curl$) and $P$ be a linear constant coefficient first order differential operator with finite dimensional kernel (for example $\nabla$).
We say that $Q$ and $P$ are exact if $Im(P)=ker (Q)$, i.e., the image of $P$ is the kernel of $Q$.
I updated my question. The original one was too vague.
My question: let the 1st order operator $P$ be given. Is there exist a 2nd order operator $Q$ such that $Q$ and $P$ are exact and there exists some sort of integration by parts formula between those two operators? i.e., $Q$ and $P$ satisfies something like:
$$ \int Qu\cdot v dx =? \int (Q^*u)\cdot (Pv)dx $$ where $u$ $v$ are both $C_c^\infty$ functions. I am aware of Laplace-Beltrami operator but it is not really what I wanted. My purpose is to find a way to have operator $Q$ on left side only, and $P$ on other side with $v$ (it is OK to have something left over with $Q$ on $u$ on right hand as well). So please feel free to suggest any possible relations.
Any possible suggestions and reference are really appreciated!