Let $u\in H^2(\Omega)$ where $\Omega$ is bounded domain in 2D.
Then is it possible to have following inequality?
\begin{equation} |u|_{H^2(\Omega)}\leq C\|\Delta u\|_{L^2(\Omega)} \end{equation}
I guess it has to be something with the elliptic regularity for Poisson equation. So, do I require $\Omega$ satisfies some properties such as Lipschitz continuous or smooth boundary?
Specifically to my case, I have B-spline space $V_h\subset H^2$ without boundary restriction and $H^2_0$ space. i.e., restriction of boundary is only piecewise polynomial.
Then is it possible to have \begin{equation} |u|_{H^2(\Omega)}\leq C\|\Delta u)\|_{L^2(\Omega)},\quad\forall u\in V_h+H^2_0(\Omega) \end{equation}
Edit
I found a theorem:
If $\partial\Omega$ is $C^2$, then \begin{equation} \|u\|_{H^2(\Omega)}\leq C(\|\Delta u\|_{L^2(\Omega)}+\|u\|_{L^2(\Omega)}) \end{equation} But in my case, the domain is square. Is there any possible way to avoid the restriction of boundary?