Propagation of fronts in a nonlinear fourth order equation

We consider a geometric motion associated with the minimization
of a curvature dependent functional, which is related to the Willmore
functional. Such a functional arises in connection with the
image segmentation problem in computer vision theory.
We show by using formal asymptotics that the geometric motion
can be approximated by the evolution of the zero
level set of the solution of a nonlinear fourth-order
equation related to the Cahn-Hilliard and Allen-Cahn equations.