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

Robert Gulliver, Yonghoi Koo

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

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 ℝn of radius ct1/3, but its complement meets a ball of a larger radius κ0t1/3. 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.

Original languageEnglish (US)
Pages (from-to)1-24
Number of pages24
JournalJournal fur die Reine und Angewandte Mathematik
Volume538
DOIs
StatePublished - 2001

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