I have Föppl–von-Karman plate equation:
$$ \Delta^2 w - (w_{xx} \varphi_{yy} + w_{yy} \varphi_{xx}-2w_{xy} \varphi_{xy}) = f\\ \Delta^2 \varphi + (w_{xx}w_{yy}-w_{xy}^2) = 0 $$
How do I write this equation in a weak form (suitable for FEM calculation)? I know how to integrate by parts $\nabla^2$ and $\Delta^2$ operators, but what should I do with von Karman bracket $[u, v]=u_{xx} v_{yy} + u_{yy} v_{xx}-2u_{xy} v_{xy}$?