When a hypersurface of Rⁿ evolves in time with normal velocity equal to its mean curvature plus a forcing term   g(x,t), the viscosity solution may be "fattened" at some moment when the hypersurface is singular.  The viscosity solution may be thought of as the union of all possible weak solutions, and fattening is the phenomenon of replacing a   hypersurface by a closed set with nonempty interior.  A specific type of geometric singularity occurs if the  hypersurface includes two smooth pieces, at the moment t = 0 when the two pieces touch each other.  If each piece is strictly convex at that moment and at that point, then we show that fattening occurs at the rate t^1/3.  That is, for small positive time, the  generalized solution contains a ball of Rⁿ of radius c t^1/3 but its complement meets a ball of a larger radius C t^1/3 . Inthis sense, the sharp rate of fattening of the viscosity solution is characterized.  The upper bound may be proved using a one-parameter family of  hypersurfaces generated by a modified "Grim Reaper."  The lower bound requires construction of a rotationally symmetric piece of hypersurface moving by mean curvature at the t^1/3 rate.  We assume that the smooth evolution of the two pieces of the  hypersurface, considered separately, do not cross each other for small positive time.

This is joint work with Yonghoi Koo.
 
 

Convenors:Klaus Ecker, Alan Pryde