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MiS Preprint
65/1999
Sharp growth rate for generalized solutions evolving by mean curvature plus a forcing term
Robert Gulliver and Yonghoi Koo
Abstract
When a hypersurface $\Sigma (t)$ 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 $\Sigma (t)$ is singular. This phenomenon corresponds to nonuniqueness of codimension-one solutions. A specific type of geometric singularity occurs if $\Sigma (t)$ 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 $t^{1/3}$ That is, for small positive time, the generalized solution contains a ball of $\mathbb(R)^n$ of radius $ct^{1/3}$, but its complement meets a ball of a larger radius $k_0 t^{1/3} $ 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 $\Sigma (t)$, considered separately, do not cross each other for small positive time.
