On mean curvature flow with forcing

Inwon Kim, Dohyun Kwon

Research output: Contribution to journalArticlepeer-review

9 Scopus citations

Abstract

This paper investigates geometric properties and well-posedness of a mean curvature flow with volume-dependent forcing. With the class of forcing which bounds the volume of the evolving set away from zero and infinity, we show that a strong version of star-shapedness is preserved over time. More precisely, it is shown that the flow preserves the ρ-reflection property, which corresponds to a quantitative Lipschitz property of the set with respect to the nearest ball. Based on this property we show that the problem is well-posed and its solutions starting with ρ-reflection property become instantly smooth. Lastly, for a model problem, we will discuss the flow’s exponential convergence to the unique equilibrium in Hausdorff topology. For the analysis, we adopt the approach developed by Feldman-Kim to combine viscosity solutions approach and variational method. The main challenge lies in the lack of comparison principle, which accompanies forcing terms that penalize small volume.

Original languageEnglish
Pages (from-to)414-455
Number of pages42
JournalCommunications in Partial Differential Equations
Volume45
Issue number5
DOIs
StatePublished - 3 May 2020

Keywords

  • Mean curvature flow
  • minimizing movements
  • moving planes method
  • star-shaped
  • viscosity solutions

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