Elliptic regularity Neumann problem in non-Lipschitz domain

Question

Let $\Omega$ be a bounded open connected domain of $\mathbb R^3$ with polyhedral, but not Lipschitz in general, boundary. Let $f\in L^2(\Omega)/ \mathbb R$. We consider the following. Find $\varphi:\Omega\to\mathbb R$ such that \begin{equation*} \begin{cases} -\Delta\varphi = f \quad\text{in}\;\Omega,\\ \frac{\partial\varphi}{\partial n}=0\quad\text{on}\;\partial\Omega, \end{cases} \end{equation*} which, in variational form, reads as follows. Find $\varphi\in H^1(\Omega)$ such that \begin{equation*} \int_{\Omega}\nabla u\cdot\nabla v = \int_{\Omega} f v\qquad\forall\ v\in H^1(\Omega). \end{equation*} Can I conclude that there exists $C>0$ such that \begin{equation*} \|{\Delta \varphi\|}_{L^2(\Omega)}\le C \|f \|_{L^2(\Omega)}? \end{equation*} In general, which is the ''smaller'' Sobolev space to which $\varphi$ belong?