Let

- $\Omega\subseteq\mathbb R^d$ be open ($d\in\mathbb N$)
- $\mathcal D:=C_c^\infty(\Omega)^d$ and $$\mathfrak D:=\left\{\phi\in\mathcal D:\nabla\cdot\phi=0\right\}$$
- $\mathcal H:=\overline{\mathfrak D}^{\langle\;\cdot\;,\;\cdot\;\rangle_{L^2(\Omega,\:\mathbb R^d)}}$ and $\mathcal V:=H_0^1(\Omega,\mathbb R^d)\cap\mathcal H$

If $$\mathfrak a(u,v):=\sum_{i=1}^d\langle\nabla u_i,\nabla v_i\rangle_{L^2(\Omega,\:\mathbb R^d)}\;\;\;\text{for }u,v\in H_0^1(\Omega,\mathbb R^d)\;,$$ then the *Dirichlet-Laplace operator* is defined to be $$\Delta_D:H_0^1(\Omega,\mathbb R^d)\to H^{-1}(\Omega,\mathbb R^d)\;,\;\;\;u\mapsto-\mathfrak a(\;\cdot\;,u)\;.$$ If $\tilde\iota:\mathcal V\to H_0^1(\Omega,\mathbb R^d)$ is the inclusion and $\tilde{\operatorname P}:H^{-1}(\Omega,\mathbb R^d)\to\mathcal V'$ is the adjoint of $\tilde\iota$, then $$A_0:\mathcal V\to\mathcal V'\;,\;\;\;u\mapsto-\tilde{\operatorname P}\Delta_D\tilde\iota u$$ is a well-defined bounded, linear and symmetric operator with $$A_0u=\left.(-\Delta_Du)\right|_{\mathcal V}\;\;\;\text{for all }u\in\mathcal V\tag 1$$ and the *Stokes operator* is defined to be the restriction $A$ of $A_0$ to $$D(A):=\left\{u\in\mathcal V:A_0u\in\mathcal H\right\}\;.\tag 2$$

Now, I've observed the following: If $u\in H_0^1(\Omega)$ is twice weak differentiable (e.g. $u\in H_0^2(\Omega)$), then $u$ has a weak Laplacian $\Delta u\in L_{\text{loc}}^1(\Omega)$ and $$(\Delta_Du)v=\langle v,\Delta u\rangle_{L^2(\Omega,\:\mathbb R^d)}\;\;\;\text{for all }v\in H_0^1(\Omega,\mathbb R^d)\;.\tag 3$$ If $\iota:\mathcal H\to L^2(\Omega,\mathbb R^d)$ is the inclusion and $\operatorname P:L^2(\Omega,\mathbb R^d)\to\mathcal H$ is the orthogonal projection, then $\iota^\ast=\operatorname P$ and hence (using $(3)$) $$(\Delta_Du)v=\langle v,\operatorname P\Delta u\rangle_{L^2(\Omega,\:\mathbb R^d)}\;\;\;\text{for all }v\in\mathcal V\;.\tag 4$$ Thus, if $u\in\mathcal V$, we obtain $$(A_0u)v=\langle v,-\operatorname P\Delta u\rangle_{L^2(\Omega,\:\mathbb R^d)}\;\;\;\text{for all }v\in\mathcal V\;.\tag 5$$ by $(1)$, i.e. $\Delta_Du\in\mathcal H$.

So, my question is: Shouldn't we be able to define the Stokes operator to be $$\tilde A:D(\tilde A)\to\mathcal H\;,\;\;\;u\mapsto-\operatorname P\Delta u$$ with $D(\tilde A)=H^2(\Omega,\mathbb R^d)\cap\mathcal V$? I've read that this isn't possible, unless $\Omega$ is regular enough, but I don't see why.