We prove the existence of a weak global in time mean curvature flow of a bounded partition of space using the method of minimizing movements. The result is extended to the case when suitable driving forces are present. We also prove some consistency results for a minimizing movement solution with smooth and viscosity solutions when the evolution starts from a partition made by a union of bounded sets at a positive distance. In addition, the motion starting from the union of convex sets at a positive distance agrees with the classical mean curvature flow and is stable with respect to the Hausdorff convergence of the initial partitions.

MSC codes

  1. mean curvature flow
  2. partitions
  3. minimizing movements

MSC codes

  1. 35D30
  2. 49J45
  3. 49J53

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Information & Authors


Published In

cover image SIAM Journal on Mathematical Analysis
SIAM Journal on Mathematical Analysis
Pages: 4117 - 4148
ISSN (online): 1095-7154


Submitted: 30 November 2017
Accepted: 2 May 2018
Published online: 24 July 2018

MSC codes

  1. mean curvature flow
  2. partitions
  3. minimizing movements

MSC codes

  1. 35D30
  2. 49J45
  3. 49J53



Shokhrukh Yu. Kholmatov

Funding Information

Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni https://doi.org/10.13039/100012740

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