Preprint 65/1999

Sharp growth rate for generalized solutions evolving by mean curvature plus a forcing term

Robert Gulliver and Yonghoi Koo

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Submission date: 09. Nov. 1999
Pages: 27
published in: Journal für die reine und angewandte Mathematik, 538 (2001), p. 1-24 
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When a hypersurface tex2html_wrap_inline13 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 tex2html_wrap_inline13 is singular. This phenomenon corresponds to nonuniqueness of codimension-one 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 tex2html_wrap_inline23 That is, for small positive time, the generalized solution contains a ball of tex2html_wrap_inline25 of radius tex2html_wrap_inline27, but its complement meets a ball of a larger radius tex2html_wrap_inline29 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 tex2html_wrap_inline31 considered separately, do not cross each other for small positive time.

24.11.2021, 02:10