Which Sobolev-Space to use to formulate weak biharmonic equation, $H^2_0$ or $H_0^1\cap H^2$? For the weak formulation of the biharmonic equation on a smooth domain $\Omega$
$$
\Delta^2u=0\;\text{in}\;\Omega\\
u=0, \nabla u\cdot \nu=0\; \text{on}\; \partial\Omega
$$
why does one take $H^2_0(\Omega)=\overline{C_c^\infty(\Omega)}^{W^{2,2}}$ as the underlying space? (i.e. $u\in H^2_0$ weak solution iff $\int \Delta u\Delta\phi=0,\;\forall \phi\in H^2_0$)
Isn't $\nabla u=0$ on $\partial\Omega$ for $u\in H^2_0(\Omega)\cap C^1(\Omega)$, which is more than $\nabla u\cdot \nu=0$?
If yes, wouldn't $H_0^1(\Omega)\cap H^2(\Omega)$ be the better choice?

Here is what I thought. I am not very sure but I am happy to discuss with you. 
The way we cast $\Delta^2u=0$ into a weak formulation, so that we could use Lax-Milgram, tells us that $H_0^2$ is a suitable space.
Suppose we have a nice solution already, then we test $\Delta^2u=0$ with a $C^\infty$ function $v$ and see what happens. 
We have 
$$\int_\Omega \Delta^2u\,v\,dx =  -\int_\Omega \nabla \Delta u \nabla v\,dx+\int_{\partial \Omega}\nabla\Delta u \cdot\nu\,v\,d\sigma(x) = \int_\Omega\Delta u\Delta v \,dx-\int_{\partial \Omega}\Delta u\nabla v\cdot\nu \,d\sigma(x)+\int_{\partial \Omega}\nabla\Delta u \cdot\nu\,v\,d\sigma(x) $$
Hence, we would like to ask $v=0$ and $\nabla v\cdot\nu=0$ on $\partial \Omega$ so that we can obtain a Bilinear operator. 
Now we have to choose our underlying space so that we could apply Lax-Milgram. Keep in mind that we need to choose a Banach space $H$, and it has to contain the condition that $v=0$ and $\nabla v\cdot\nu=0$ on $\partial \Omega$. Certainly $H_0^1\cap H^2(\Omega)$ will not work because it can not confirm that $\nabla v\cdot \nu\equiv 0$. Hence the only natural space left for us is $H_0^2(\Omega)$.
I am really not sure that $\{v\in H^2, T[u]=0\text{ and }T[\nabla u]\cdot \nu=0\}$ is a Banach space. 
