# Regular weak solution is classical solution biharmonic problem

I'm trying to show that if $$u \in C^4(\Omega)\cap C^1(\overline{\Omega})$$ satisfies the weak formulation of the biharmonic problem $$\int_\Omega \Delta u\Delta \varphi = \int_\Omega f \varphi \ \ \ \ \ \forall \varphi \in H_0^2(\Omega)$$ then $$u$$ is a classical solution of $$\Delta^2 u = f \ \text{ in } \Omega \\ u = \frac{\partial u}{\partial \eta} = 0 \ \text{ in } \partial \Omega$$

Here I'm assuming the domain is well behaved (open, bounded and with smooth boundary) and $$f \in L^2$$.

My doubt is that i get a bit lost still in what aproaches i can use, following this answer i was thinking of doing something like

$$0 = \int \Delta u \Delta \varphi - f \varphi \\ =\int_\partial \nabla \varphi \Delta u \cdot\eta \ - \int_\partial \varphi \nabla(\Delta u) \cdot \eta \ + \int (\Delta^2 u- f)\varphi$$

and from there i could conclude that $$\Delta^2 u = f$$ but I'd have to handle two integrals for the boundary conditions and i can't conclude from there, am I missing something?

Actually forgot a vital piece of information, $$u \in H_0^2$$, so therefore the boundary conditions arise from the space where the weak solution lies.