  We consider a mixed finite element method for a biharmonic equation with
clamped boundary conditions based on biorthogonal systems with weakly imposed
Dirichlet boundary condition. We show that the weak imposition of the boundary
condition arising from a natural minimisation formulation allows to get an
optimal a priori error estimate for the finite element scheme improving the
existing error estimate for such a formulation without weakly imposed Dirichlet
boundary condition. We also briefly outline the algebraic formulation arising
from the finite element method.
