Abstract

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.

Keywords

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

MSC codes

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

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References

1.
F. Almgren, J. E. Taylor, and L. Wang, Curvature-driven flows: A variational approach, SIAM J. Control Optim., 31 (1993), pp. 387--438.
2.
L. Ambrosio, Movimenti minimizzanti, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 113 (1995), pp. 191--246.
3.
L. Ambrosio, N. Fusco, and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, New York, 2000.
4.
L. Ambrosio, N. Gigli, and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Birkhäuser-Verlag, Basel, Switzerland, 2008.
5.
J. Ball, D. Kinderlehrer, P. Podio-Guidugli, and M. Slemrod, Fundamental Contributions to the Continuum Theory of Evolving Phase Interfaces in Solids, Springer-Verlag, Berlin, 1999.
6.
G. Bellettini, Lecture Notes on Mean Curvature Flow: Barriers and Singular Perturbations, Appunti. Sc. Norm. Super. Pisa (N.S.) 12, Edizioni della Normale, Pisa, Italy, 2013.
7.
G. Bellettini, V. Caselles, A. Chambolle, and M. Novaga, Crystalline mean curvature flow of convex sets, Arch. Ration. Mech. Anal., 179 (2006), pp. 109--152.
8.
G. Bellettini and S. Kholmatov, Minimizing movements for mean curvature flow of droplets with prescribed contact angle, J. Math. Pures Appl., to appear.
9.
K. Brakke, The Motion of a Surface by Its Mean Curvature, Math. Notes 20, Princeton University Press, Princeton, 1978.
10.
D. Caraballo, A Variational Scheme for the Evolution of Polycrystals by Curvature, Ph.D. thesis, Princeton University, Princeton, NJ, 1996.
11.
A. Chambolle, An algorithm for mean curvature motion, Interfaces Free Bound., 6 (2004), pp. 195--218.
12.
A. Chambolle, M. Morini, and M. Ponsiglione, Nonlocal curvature flows, Arch. Ration. Mech. Anal., 218 (2015), pp. 1263--1329.
13.
A. Chambolle and M. Novaga, Implicit time discretization of the mean curvature flow with a discontinuous forcing term, Interfaces Free Bound., 10 (2008), pp. 283--300.
14.
T. Colding and W. P. Minicozzi II, A Course in Minimal Surfaces, Grad. Stud. Math. 12, American Mathematical Society, Providence, RI, 2011.
15.
E. De Giorgi, Sulla proprietà isoperimetrica dell'ipersfera, nella classe degli insiemi aventi frontiera orientata di misura finita, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Nat. Sez. I (8), 5 (1958), pp. 33--44.
16.
E. De Giorgi, Complementi alla teoria della misura $(n-1)$-dimensionale in uno spazio $n$ dimensionale, Sem. Mat. Scuola Norm. Sup. Pisa, 1960--61, Editrice Tecnico Scientifica, Pisa, Italy, 1961.
17.
E. De Giorgi, New problems on minimizing movements, in Boundary Value Problems for Partial Differential Equations and Applications, Masson, Paris, 1993, pp. 81--98.
18.
E. De Giorgi, Movimenti di partizioni, in Variational Methods for Discontinuous Structures, R. Serapioni, F. Tomarelli, eds., Progr. Nonlinear Differential Equations Appl. 25, Birkhäuser, Basel, Switzerland, 1996, pp. 1--5.
19.
D. Depner, H. Garcke, and Y. Kohsaka, Mean curvature flow with triple junctions in higher space dimensions, Arch. Ration. Mech. Anal., 211 (2014), pp. 301--334.
20.
K. Ecker, Regularity Theory for Mean Curvature Flow, Birkhäuser, Basel, Switzerland, 2004.
21.
S. Esedoḡlu and F. Otto, Threshold dynamics for networks with arbitrary surface tensions, Comm. Pure Appl. Math., 68 (2015), pp. 808--864.
22.
L. Evans, H. Soner, and P. Souganidis, Phase transitions and generalized motion by mean curvature, Comm. Pure Appl. Math., 45 (1992), pp. 1097--1123.
23.
F. Ferrari, B. Franchi, and G. Lu, On a relative Alexandrov-Fenchel inequality for convex bodies in Euclidean spaces, Forum Math., 18 (2006), pp. 907--921.
24.
A. Freire, Mean curvature motion of graphs with constant contact angle at a free boundary, Anal. PDE, 3 (2010), pp. 359--407.
25.
A. Freire, Mean curvature motion of triple junctions of graphs in two dimensions, Comm. Partial Differential Equations, 35 (2010), pp. 302--327.
26.
M. Gage and R. Hamilton, The heat equation shrinking convex plane curves, J. Differential Geom., 23 (1986), pp. 69--95.
27.
Y. Giga, Surface Evolution Equations, Birkhäuser, Basel, Switzerland, 2006.
28.
E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhäuser, Basel, Switzerland, 1984.
29.
G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom., 20 (1984), pp. 237--266.
30.
T. Ilmanen, Elliptic regularization and partial regularity for motion by mean curvature, Mem. Amer. Math. Soc., 108 (1994), 520.
31.
L. Kim and Y. Tonegawa, On the mean curvature flow of grain boundaries, Ann. Inst. Fourier (Grenoble), 67 (2017), pp. 43--142.
32.
D. Kinderlehrer and C. Liu, Evolution of grain boundaries, Math. Models Methods Appl. Sci., 11 (2001), pp. 713--729.
33.
T. Laux and F. Otto, Convergence of the thresholding scheme for multi-phase mean-curvature flow, Calc. Var. Partial Differential Equations, 55 (2016), pp. 55--129.
34.
T. Laux and D. Swartz, Convergence of thresholding schemes incorporating bulk effects, Interfaces Free Bound., 19 (2017), pp. 273--304.
35.
G. Leonardi and I. Tamanini, Metric spaces of partitions, and caccioppoli partitions, Adv. Math. Sci. Appl., 12 (2002), pp. 725--753.
36.
S. Luckhaus and T. Sturzenhecker, Implicit time discretization for the mean curvature flow equation, Calc. Var. Partial Differential Equations, 3 (1995), pp. 253--271.
37.
F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems. An Introduction to Geometric Measure Theory, Cambridge University Press, Cambridge, 2012.
38.
C. Mantegazza, Lecture Notes on Mean Curvature Flow, Birkhäuser, Basel, Switzerland, 2011.
39.
C. Mantegazza, M. Novaga, A. Pluda, and F. Schulze, Evolution of Networks with Multiple Junctions, preprint, https://arxiv.org/abs/1611.08254 [Math. DG], 2016.
40.
B. Merriman, J. Bence, and S. Osher, Diffusion Generated Motion by Mean Curvature, Manuscript, Department of Mathematics, University of California, Los Angeles, CA, 1992.
41.
B. Merriman, J. Bence, and S. Osher, Motion of multiple junctions: A level set approach, J. Comput. Phys., 112 (1994), pp. 334--363.

Information & Authors

Information

Published In

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

History

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

Keywords

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

MSC codes

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

Authors

Affiliations

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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