Weak formulation of a system of biharmonic pdes Consider the system of pdes for functions $u,v$
$$\begin{cases} a_0\Delta ^2u+a_1u_{xx}v_{xx}+a_2u_{xy}v_{xy}+a_3u_{yy}v_{yy}  = f\\
b_0\Delta ^2v+b_1u_{xx}v_{xx}+b_2u_{xy}v_{xy}+b_3u_{yy}v_{yy} = g 
\end{cases}$$
subject to homogeneous Dirichlet boundary conditions over an open bounded connected domain $(x,y) \in \Omega $, where $a_1,a_2,a_3, b_1,b_2,b_3\in L^{2}(\Omega)$, and $a_0>0, b_0>0$ are both constants, $\Delta ^2: =\left(\frac{\partial^2}{\partial x^2}+ \frac{\partial^2}{\partial y^2}\right)^2=\frac{\partial^4}{\partial x^4}+2 \frac{\partial^2}{\partial x\partial y}+\frac{\partial^4}{\partial y^4}$ is the biharmonic operator.
Formulate a weak definition of the solution. What can you say about the existence and uniqueness of $u,v$ based on your weak formulation?
 A: Followed by 'Quickbeam2k1's comment, following is my weak formulation.
Assume $u,v \in H^2(\Omega)$
Let $(U,V)\in H^2(\Omega)\times H^2(\Omega),$ Multiplying the first and second equation by $U$ and $V$ respectively (how does one write this multiplication abstractly in terms of an operator $L$?) , then employing Greens's identity (integrate by parts) to the biharmonic terms to obtain  $$\begin{cases} a_0\int_{\Omega}\nabla(\Delta u)\cdot \nabla U\mathrm{d}x-a_0\int_{\partial\Omega}\left(\Delta u \frac{\partial U}{\partial n}-U\frac{\partial \Delta U}{\partial n}\right)\mathrm{d}S+\int_{\Omega}C\cdot U\mathrm{d}x = \int_{\Omega}fU\mathrm{d}x\\
b_0\int_{\Omega}\nabla(\Delta v)\cdot \nabla V\mathrm{d}x-b_0\int_{\partial\Omega}\left(\Delta v \frac{\partial V}{\partial n}-V\frac{\partial \Delta V}{\partial n}\right)\mathrm{d}S+\int_{\Omega}D\cdot V\mathrm{d}x  =\int_{\Omega}gV\mathrm{d}x
\end{cases}$$
Where $\begin{cases}C:= a_1u_{xx}v_{xx}+a_2u_{xy}v_{xy}+a_3u_{yy}v_{yy}\\D: = b_1u_{xx}v_{xx}+b_2u_{xy}v_{xy}+b_3u_{yy}v_{yy} \end{cases}$,$$n=\mathrm{the~ outer~ unit~ normal}
$$. 
Is my formulation correct? If so, how to identify the bilinear operator $B$ for Lax-Milgram?
