Suppose $\Omega$ is a bounded smooth domain. If $u \in H^1(\Omega)$ is such that $\Delta u \in L^2(\Omega)^* \subset H^1(\Omega)^*$, does it follow that $u \in H^2(\Omega)$?
By $-\Delta u$ I mean the weak Laplacian, which usually is defined as the object in $H^1(\Omega)^*$ such that $$\langle -\Delta u , v \rangle = \int_\Omega \nabla u \nabla v$$ for every $v \in H^1(\Omega)$.
I cannot identify $L^2(\Omega)$ and $L^2(\Omega)^*$, for certain reasons. If I could do so, then the answer is immediate from elliptic regularity.