Flow by mean curvature: the sharp rate of fattening

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


When a hypersurface tex2html_wr
<p>ap_inline13 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

=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline13"

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

tex2html_wrap_inline13 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 te

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

positive time, the generalized solution contains a ball of

HEIGHT=13 ALIGN=BOTTOM ALT="tex2html_wrap_inline25"


of radius tex2html_wrap_inline2

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


radius tex2html_wrap_inline29

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 tex2html_wrap_inl
<p>ine31 considered separately, do not c

ross each other

for small positive time.

Katharina Matschke

MPI for Mathematics in the Sciences Contact via Mail