I've been having some trouble proving this, so any help or suggestions would be great. Thanks in advance!
Show using a Green's formula that, for any $u \in H^2(\Omega)$ satisfying $$u = \frac{\partial u}{\partial n} = 0$$ we have $$\int_\Omega |\Delta u|^2 \, dx \, dy = \int_\Omega \left[(u_{xx})^2 + (u_{yy})^2 - (u_{xy})^2 + (u_{yx})^2 \right] \, dx \, dy$$
My first instinct is to use integration by parts on the left hand side, but I'm a little confused because it says $u \in H^2(\Omega)$, so doesn't that mean I can't have three derivatives on $u$?
By the way, we can assume that $\Omega$ has sufficiently smooth boundary. Also, $\frac{\partial}{\partial n}$ denotes differentiation in the outward normal direction to the boundary $\Gamma$.
