On singular perturbations of some partial differential equations

It is known that mean curvature flow of the boundary of a smooth bounded open subset of R n develops singularities at finite time: the simplest example is when n = 2, for which the singularity is the disappearence of the curve, i.e., when the curve reduces to a point [5]. It is therefore reasonable to regularize the problem with a sequence of problems which admit a global solution and then to try to pass to the limit as the regularization parameter converges to zero. One possible regularization consists in adding, at the level of the energy functionals, an higher order term.