Ciarlet-Raviart for Biharmonic equation, Underconstrained system

Hi there. I am using FreeFEM++ to solve a biharmonic equation using Ciarlet-Raviart formulation. I want to know what happens if “solve” has an underconstrained system?

I am asking because I disregard the fact that one of the test functions should be in H^1_0, and I am not getting what I am expecting. The formulation also has two equalities and I provide them as a single one to “solve”.

I will really appreciate your help.

I guess you could post your code but if you are looking into this anyway it
may help to just write out the weak formulation and see what happens.
You can formulate a simple problem on a small square grid mesh
with a FF “varf” and dump the matrix and rhs and
see if it makes sense.