Flow by mean curvature: the sharp rate of fattening

  • Robert Gulliver (University Minnesota + MPI MiS, Leipzig)
A3 02 (Seminar room)


When a hypersurface Σ(t) evolves with normal velocity equal to its mean curvature plus a forcing term g(x,t), the generalized (viscosity) solution may be "fattened" at some moment when Σ(t) is singular. This phenomenon corresponds to nonuniqueness of codimension-one solutions. A specific type of geometric singularity occurs if Σ(t) 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 t1/3. That is, for small positive time, the generalized solution contains a ball of IRn of radius ct1/3, but its complement meets a ball of a larger radius k0t1/3. In this sense, the sharp rate of fattening of the generalized solution is characterized. We assume that the smooth evolution of the two pieces of Σ(t), considered separately, do not cross each other for small positive time.

Katharina Matschke

MPI for Mathematics in the Sciences Contact via Mail