Flow by mean curvature: the sharp rate of fattening

When a hypersurface evolves with normal veloc

ity equal

to its mean curvature plus a forcing term g(x,t), th

e generalized (viscosity)

solution may be "fattened" at some moment when

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SRC="gulliver/prepr6599-img1.gif"> is

singular. This phenomenon corresponds to nonuniqueness of codimension-on

e solutions. A specific type of geometric singularity occurs if

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

SRC="gulliver/prepr6599-img2.gif"> That is, for small

positive time, the generalized solution contains a ball of

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of radius

SRC="gulliver/prepr6599-img4.gif">, but its complement meets a ball of a

larger

radius

SRC="gulliver/prepr6599-img5.gif"> In this sense, the sharp rate of fatt

ening of

the generalized

solution is characterized. We assume that the smooth evolution of the

two pieces of considered separately, do not c

ross each other

for small positive time.