Hostname: page-component-cb9f654ff-5jtmz Total loading time: 0 Render date: 2025-09-02T12:11:51.594Z Has data issue: false hasContentIssue false
  • English
  • Français

Minimizing movements for forced anisotropic mean curvature flow of partitions with mobilities

Part of: Global differential geometry Existence theories

Published online by Cambridge University Press:  17 August 2020

Giovanni Bellettini ,
Antonin Chambolle  and
Shokhrukh Kholmatov
Giovanni Bellettini
Affiliation:
University of Siena, via Roma 56, 53100, Siena, Italy International Centre for Theoretical Physics (ICTP), Strada Costiera 11, 34151, Trieste, Italy (bellettini@diism.unisi.it)
Antonin Chambolle
Affiliation:
CMAP, Ecole Polytechnique, CNRS, 91128, Palaiseau Cedex, France (antonin.chambolle@cmap.polytechnique.fr)
Shokhrukh Kholmatov
Affiliation:
University of Vienna, Oskar-Morgenstern-Platz 1, 1090, Vienna, Austria (shokhrukh.kholmatov@univie.ac.at)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Under suitable assumptions on the family of anisotropies, we prove the existence of a weak global 1/(n+1)-Hölder continuous in time mean curvature flow with mobilities of a bounded anisotropic partition in any dimension using the method of minimizing movements. The result is extended to the case when suitable driving forces are present. We improve the Hölder exponent to 1/2 in the case of partitions with the same anisotropy and the same mobility and provide a weak comparison result in this setting for a weak anisotropic mean curvature flow of a partition and an anisotropic mean curvature two-phase flow.

Keywords

mean curvature flowpartitionsminimizing movementsforcinganisotropymobility

MSC classification

Primary: 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.)
Secondary: 49J45: Methods involving semicontinuity and convergence; relaxation 49J53: Set-valued and variational analysis

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 4 , August 2021 , pp. 1135 - 1170
Check for updates
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
Copyright © The Author(s) 2020. Published by Cambridge University Press

References

Almgren, F., Taylor, J. and Wang, L.. Curvature-driven flows: a variational approach. SIAM J. Control Optim. 31 (1993), 387–438. (doi: 10.1137/0331020).CrossRefGoogle Scholar
Almgren, F. and Taylor, J. E.. Flat flow is motion by crystalline curvature for curves with crystalline energies. J. Differential Geom. 42 (1995), 1–22. (doi:10.4310/jdg/1214457030).CrossRefGoogle Scholar
Ambrosio, L.. Movimenti minimizzanti. Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. 19 (1995), 191–246. (MR1387558).Google Scholar
Ambrosio, L., Fusco, N. and Pallara, D.. Functions of bounded variation and free biscontinuity problems (New York: Oxford University Press, 2000).Google Scholar
Angenent, S. B. and Gurtin, M. E.. Multiphase thermomechanics with interfacial structure 2. Evolution of an isothermal interface. Arch. Rational Mech. Anal. 108 (1989), 323–391. (doi: 10.1007/BF01041068).CrossRefGoogle Scholar
Bellettini, G., Novaga, M. and Paolini, M.. On a crystalline variational problem, Part I: first variation and global L ∞ regularity. Arch. Rational Mech. Anal. 157 (2001), 165–191. (doi: 10.1007/s002050010127).CrossRefGoogle Scholar
Bellettini, G., Novaga, M. and Paolini, M.. On a crystalline variational problem, part II: BV-regularity and structure of minimizers on facets. Arch. Rational Mech. Anal. 157 (2001), 193–217. (doi: 10.1007/s002050100126).CrossRefGoogle Scholar
Bellettini, G., Riey, G. and Novaga, M.. First variation of anisotropic energies and crystalline mean curvature for partitions. Interfaces Free Bound. 5 (2003), 331–356. (doi: 10.4171/IFB/82).CrossRefGoogle Scholar
Bellettini, G.. Anisotropic and Crystalline Mean Curvature Flow. In A Sampler of Riemann-Finsler Geometry (eds. Bao, D., Bryant, R. L., Chern, S.-S., Shen, Z.), Mathematical Sciences Research Institute Publications, vol. 50, pp. 49–83 (Cambridge Univ. Press, 2004).Google Scholar
Bellettini, G., Caselles, V., Chambolle, A. and Novaga, M.. Crystalline mean curvature flow of convex sets. Arch. Rational Mech. Anal. 179 (2005), 109–152. (doi: 10.1007/s00205-005-0387-0).CrossRefGoogle Scholar
Bellettini, G., Chermisi, M. and Novaga, M.. Crystalline curvature flow of planar networks. Interfaces Free Bound. 8 (2006), 481–521. (doi: 10.4171/IFB/152).CrossRefGoogle Scholar
Bellettini, G.. Lecture notes on mean curvature flow, barriers and singular perturbations. vol. 12, (Pisa: Publications of the Scuola Normale Superiore di Pisa, 2013).CrossRefGoogle Scholar
Bellettini, G., Kholmatov, S. Y.. Minimizing movements for mean curvature flow of droplets with prescribed contact angle. J. Math. Pures Appl. 117 (2018), 1–58. (doi: 10.1016/j.matpur.2018.06.003).CrossRefGoogle Scholar
Bellettini, G., Kholmatov, S. Y.. Minimizing movements for mean curvature flow of partitions. SIAM J. Math. Anal. 50 (2018), 4117–4148. (doi: 10.1137/17M1159294).CrossRefGoogle Scholar
Brakke, K.. The motion of a surface by its mean curvature Math. Notes. vol. 20, (Princeton: Princeton University Press, 1978).Google Scholar
Chambolle, A.. An algorithm for mean curvature motion. Interfaces Free Bound. 6 (2004), 195–218. (doi: 10.4171/IFB/97).CrossRefGoogle Scholar
Chambolle, A., Morini, M., Novaga, M. and Ponsiglione, M.. Existence and uniqueness for anisotropic and crystalline mean curvature flows. J. Amer. Math. Soc. 32 (2019), 779–824. (doi: 10.1090/jams/919).CrossRefGoogle Scholar
Chambolle, A., Morini, M. and Ponsiglione, M.. Nonlocal curvature flows. Arch. Rational Mech. Anal. 218 (2015), 1263–1329. (doi: 10.1007/s00205-015-0880-z).CrossRefGoogle Scholar
Chambolle, A., Morini, M. and Ponsiglione, M.. Existence and uniqueness for a crystalline mean curvature flow. Comm. Pure Appl. Math. 70 (2017), 1084–1114. (doi: 10.1002/cpa.21668).CrossRefGoogle Scholar
Chambolle, A., Morini, M., Novaga, M. and Ponsiglione, M.. Generalized crystalline evolutions as limits of flows with smooth anisotropies. Analysis PDE 12 (2019), 789–813. (doi: 10.2140/apde.2019.12.789).CrossRefGoogle Scholar
De Giorgi, E.. New problems on minimizing movements. Boundary value problems for PDEs and applications. RMA Res. Notes Appl. Math. 29 (1993), 81–98, Masson, Paris.Google Scholar
De Giorgi, E.. Movimenti di partizioni. In Variational methods for free discontinuity structures, (eds. Serapioni, R., Tomarelli, F.), vol. 25, pp. 1–5. (Basel: Birkhäuser, 1996). (doi: 10.1007/978-3-0348-9244-5-1).Google Scholar
Depner, D., Garcke, H. and Kohsaka, Y.. Mean curvature flow with triple junctions in higher space dimensions. Arch. Rational Mech. Anal. 211 (2014), 301–334. (doi: 10.1007/s00205-013-0668-y).CrossRefGoogle Scholar
Evans, L., Soner, H. and Souganidis, P.. Phase transitions and generalized motion by mean curvature. Comm. Pure Appl. Math. 45 (1992), 1097–1123. (doi: 10.1002/cpa.3160450903).CrossRefGoogle Scholar
Freire, A.. Mean curvature motion of graphs with constant contact angle at a free boundary. Anal. PDE 3 (2010), 359–407. (doi: 10.2140/apde.2010.3.359).CrossRefGoogle Scholar
Freire, A.. Mean curvature motion of triple junctions of graphs in two dimensions. Commun. Partial Differ. Equ. 35 (2010), 302–327. (doi: 10.1080/03605300903419775).CrossRefGoogle Scholar
Giga, Y. and Gurtin, M. E.. A comparison theorem for crystalline evolutions in the plane. Quart. Appl. Math. 54 (1996), 727–737. (MathSciNet: MR1417236).CrossRefGoogle Scholar
Giga, M.-H. and Giga, Y.. Crystalline and level set flow – convergence of a crystalline algorithm for a general anisotropic curvature flow in the plane. In Proc. Free Boundary Problems, Theory and Applications, (ed. Kenmochi, N.), Gakuto International Series, Math. Sci. Appl., vol. 13, pp. 64–79 (Japan: Chiba, 2000).Google Scholar
Giga, Y., Paolini, M. and Rybka, P.. On the motion by singular interfacial energy. Japan J. Indust. Appl. Math. 18 (2001), 231–248. (doi:10.1007/BF03168572).CrossRefGoogle Scholar
Giga, Y.. Surface evolution equations (Basel: Birkhäuser, 2006).Google Scholar
Giga, Y. and Požár, N.. A level set crystalline mean curvature flow of surfaces. Adv. Differ. Equ. 21 (2016), 631–698. (MathSciNet: MR3493931).Google Scholar
Giga, Y. and Požár, N.. Approximation of general facets by regular facets with respect to anisotropic total variation energies and its application to crystalline mean curvature flow. Comm. Pure Appl. Math. 71 (2018), 1461–1491. (doi: 10.1002/cpa.21752).CrossRefGoogle Scholar
Grayson, M. A.. A short note on the evolution of a surface by its mean curvature. Duke Math. J. 58 (1989), 555–558. (doi: 10.1215/S0012-7094-89-05825-0).CrossRefGoogle Scholar
Huisken, G.. Asymptotic behaviour for singularities of the mean curvature flow. J. Differ. Geom. 31 (1990), 285–299. (doi: 10.4310/jdg/1214444099).CrossRefGoogle Scholar
Huisken, G.. Local and global behaviour of hypersurfaces moving by mean curvature. Proc. Sympos. Pure Math. 54 (1993), 175–191. (MathSciNet: MR1216584).CrossRefGoogle Scholar
Huisken, G.. A distance comparison principle for evolving curves. Asian J. Math. 2 (1998), 127–133. (doi: 10.4310/AJM.1998.v2.n1.a2).CrossRefGoogle Scholar
Ilmanen, T.. Elliptic regularization and partial regularity for motion by mean curvature Mem. Amer. Math. Soc. 108 (1994) AMS.Google Scholar
Ilmanen, T., Neves, A. and Schulze, F.. On short time existence for the planar network flow. J. Differ. Geom. 111 (2019), 39–89. (doi:10.4310/jdg/1547607687).CrossRefGoogle Scholar
Kim, L. and Tonegawa, Y.. On the mean curvature flow of grain boundaries. Ann. Inst. Fourier (Grenoble) 67 (2017), 43–142. (doi: 10.5802/aif.3077).CrossRefGoogle Scholar
Kinderlehrer, D. and Liu, C.. Evolution of grain boundaries. Math. Models Methods Appl. Sci. 11 (2001), 713–729. (doi: 10.1142/S0218202501001069).CrossRefGoogle Scholar
Laux, T. and Otto, F.. Convergence of the thresholding scheme for multi-phase mean curvature flow. Calc. Var. Partial Differ. Equ. 55 (2016), 55–129. (doi: 10.1007/s00526-016-1053-0).CrossRefGoogle Scholar
Luckhaus, S. and Sturzenhecker, T.. Implicit time discretization for the mean curvature flow equation. Calc. Var. Partial Differ. Equ. 3 (1995), 253–271. (doi: 10.1007/BF01205007).CrossRefGoogle Scholar
Maggi, F.. Sets of finite perimeter and geometric variational problems. An introduction to geometric measure theory (Cambridge: Cambridge University Press, 2012).CrossRefGoogle Scholar
Mantegazza, C.. Lecture notes on mean curvature flow. (Basel: Birkhäuser, 2011).CrossRefGoogle Scholar
Mantegazza, C., Novaga, M., Pluda, A. and Schulze, F.. Evolution of networks with multiple junctions. arXiv:1611.08254 [math.DG].Google Scholar
Merriman, B., Bence, J. and Osher, S.. Diffusion generated motion by mean curvature (Los Angeles, CA: Manuscript, Department of Mathematics, University of California 1992).Google Scholar
Mugnai, L. and Röger, M.. The Allen-Cahn action functional in higher dimensions. Indiana Univ. Math. J. 10 (2008), 45–78. (doi: 10.4171/IFB/179).Google Scholar
Mugnai, L., Seis, C. and Spadaro, E.. Global solutions to the volume-preserving mean-curvature flow. Calc. Var. 55 (2016), Article number 18. (doi: /10.1007/s00526-015-0943-x).CrossRefGoogle Scholar
Mullins, W.. Two-dimensional motion of idealized grain boundaries. J. Appl. Phys. 27 (1956), 900–904. (doi: 10.1007/978-3-642-59938-5-3).CrossRefGoogle Scholar
Schulze, F. and White, B.. A local regularity theorem for mean curvature flow with triple edges. J. Reine Angew. Math. 758 (2020), 281–305. (doi: 10.1515/crelle-2017-0044).CrossRefGoogle Scholar
Taylor, J. E., Cahn, J. W. and Handwerker, C. A.. Overview No. 98 I - Geometric models of crystal growth. Acta Metall. Mater. 40 (1992), 1443–1474. (doi: 10.1016/0956-7151(92)90090-2).CrossRefGoogle Scholar
Taylor, J. E.. Motion of curves by crystalline curvature, including triple junctions and boundary points. Differ Geom., Proc. Sympos. Pure Math. 54 (1993), 417–438. (doi: 10.1023/A:1004523005442).CrossRefGoogle Scholar
Taylor, J. E.. A variational approach to crystalline triple-junction motion. J. Statist. Phys. 95 (1999), 1221–1244. (doi: 10.1023/A:1004523005442).CrossRefGoogle Scholar
Tonegawa, Y.. Brakke's mean curvature flow. Springer briefs in bathematics (Singapore: Springer, 2019).CrossRefGoogle Scholar