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?

I realized that $u=0$ on $\partial\Omega$ and $\nabla u\cdot \nu=0$ imply $\nabla u=0$ on $\partial\Omega$ for smooth $u$. So taking $H^2_0$ as space where we seek the weak solution, i.e. intuitively requiring $\nabla u=0$ on $\partial\Omega$, is NOT more than $\nabla u\cdot \nu =0$
