Given the problem, $f \in L^2(\Omega)$, $a$ is continuous, coercive (elliptic) and bilinear, $L$ is continuous suppose this problem has a unique solution
$$\begin{cases} u \in V, \\ a(u,v) = L(v), \forall v \in V \end{cases}$$
my question is about working backwards to classical solutions with elliptic regularity theorem. So by setting $V=H^1_0(\Omega)$ where $\Omega$ is a regular, bounded open set of $\mathbb{R}^d$. If $u$ is a solution then $u \in H^2_{\text{loc}}(\Omega)$ by elliptic regularity theorem, and we can infer that the PDE satisfied is
$$-\Delta u = f $$
is there anything to be said about the boundary here? What if $V=H^1(\Omega)$, can elliptic regularity can be used again to show $u \in L^2_{\text{loc}}(\Omega)$? How would one find out any information about the boundary?