Define $Du= -\nabla \cdot (a(x)\nabla u)$ where $a$ is a smooth function which is away from $0$ and bounded.
Suppose I have $u \in H^1(\Omega)$ on a smooth bounded domain and I also know that $Du \in L^2(\Omega)$.
Does this imply that $\Delta u \in L^2(\Omega)$ and $u \in H^2(\Omega)$?
For the Laplacian case: we get immediately that the Laplacian is in $L^2$ and then elliptic regularity theory gives $H^2$ for $u$. But this case, I don't know????