Long time behavior of discrete volume preserving mean curvature flows

References

[1] E. Acerbi, N. Fusco and M. Morini, Minimality via second variation for a nonlocal isoperimetric problem, Comm. Math. Phys. 322 (2013), no. 2, 515–557. 10.1007/s00220-013-1733-ySearch in Google Scholar

[2] F. Almgren, J. E. Taylor and L. Wang, Curvature-driven flows: A variational approach, SIAM J. Control Optim. 31 (1993), no. 2, 387–438. 10.1137/0331020Search in Google Scholar

[3] T. Aubin, Some nonlinear problems in Riemannian geometry, Springer Monogr. Math., Springer, Berlin 1998. 10.1007/978-3-662-13006-3Search in Google Scholar

[4] G. Bellettini, V. Caselles, A. Chambolle and M. Novaga, The volume preserving crystalline mean curvature flow of convex sets in ℝN, J. Math. Pures Appl. (9) 92 (2009), no. 5, 499–527. 10.1016/j.matpur.2009.05.016Search in Google Scholar

[5] W. Carter, A. Roosen, J. Cahn and J. Taylor, Shape evolution by surface diffusion and surface attachment limited kinetics on completely faceted surfaces, Acta Metall. Mater. 43 (1995), 4309–4323. 10.1016/0956-7151(95)00134-HSearch in Google Scholar

[6] G. Ciraolo and F. Maggi, On the shape of compact hypersurfaces with almost-constant mean curvature, Comm. Pure Appl. Math. 70 (2017), no. 4, 665–716. 10.1002/cpa.21683Search in Google Scholar

[7] G. Ciraolo and L. Vezzoni, A sharp quantitative version of Alexandrov’s theorem via the method of moving planes, J. Eur. Math. Soc. (JEMS) 20 (2018), no. 2, 261–299. 10.4171/JEMS/766Search in Google Scholar

[8] R. Cristoferi, On periodic critical points and local minimizers of the Ohta–Kawasaki functional, Nonlinear Anal. 168 (2018), 81–109. 10.1016/j.na.2017.11.004Search in Google Scholar

[9] M. G. Delgadino and F. Maggi, Alexandrov’s theorem revisited, Anal. PDE 12 (2019), no. 6, 1613–1642. 10.2140/apde.2019.12.1613Search in Google Scholar

[10] M. G. Delgadino, F. Maggi, C. Mihaila and R. Neumayer, Bubbling with L2-almost constant mean curvature and an Alexandrov-type theorem for crystals, Arch. Ration. Mech. Anal. 230 (2018), no. 3, 1131–1177. 10.1007/s00205-018-1267-8Search in Google Scholar

[11] J. Escher and G. Simonett, The volume preserving mean curvature flow near spheres, Proc. Amer. Math. Soc. 126 (1998), no. 9, 2789–2796. 10.1090/S0002-9939-98-04727-3Search in Google Scholar

[12] L. Esposito and N. Fusco, A remark on a free interface problem with volume constraint, J. Convex Anal. 18 (2011), no. 2, 417–426. Search in Google Scholar

[13] M. Giaquinta and L. Martinazzi, An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs, 2nd ed., Appunti. Sc. Norm. Super. Pisa (N. S.) 11, Edizioni della Normale, Pisa 2012. 10.1007/978-88-7642-443-4Search in Google Scholar

[14] G. Huisken, The volume preserving mean curvature flow, J. reine angew. Math. 382 (1987), 35–48. 10.1515/crll.1987.382.35Search in Google Scholar

[15] I. Kim and D. Kwon, Volume preserving mean curvature flow for star-shaped sets, Calc. Var. Partial Differential Equations 59 (2020), no. 2, Paper No. 81. 10.1007/s00526-020-01738-0Search in Google Scholar

[16] B. Krummel and F. Maggi, Isoperimetry with upper mean curvature bounds and sharp stability estimates, Calc. Var. Partial Differential Equations 56 (2017), no. 2, Paper No. 53. 10.1007/s00526-017-1139-3Search in Google Scholar

[17] S. Luckhaus and T. Sturzenhecker, Implicit time discretization for the mean curvature flow equation, Calc. Var. Partial Differential Equations 3 (1995), no. 2, 253–271. 10.1007/BF01205007Search in Google Scholar

[18] F. Maggi, Sets of finite perimeter and geometric variational problems. An introduction to geometric measure theory, Cambridge Stud. Adv. Math. 135, Cambridge University, Cambridge 2012. 10.1017/CBO9781139108133Search in Google Scholar

[19] U. F. Mayer, A singular example for the averaged mean curvature flow, Exp. Math. 10 (2001), no. 1, 103–107. 10.1080/10586458.2001.10504432Search in Google Scholar

[20] U. F. Mayer and G. Simonett, Self-intersections for the surface diffusion and the volume-preserving mean curvature flow, Differential Integral Equations 13 (2000), no. 7–9, 1189–1199. 10.57262/die/1356061216Search in Google Scholar

[21] L. Mugnai and C. Seis, On the coarsening rates for attachment-limited kinetics, SIAM J. Math. Anal. 45 (2013), no. 1, 324–344. 10.1137/120865197Search in Google Scholar

[22] L. Mugnai, C. Seis and E. Spadaro, Global solutions to the volume-preserving mean-curvature flow, Calc. Var. Partial Differential Equations 55 (2016), no. 1, Article ID 18. 10.1007/s00526-015-0943-xSearch in Google Scholar

[23] J. Niinikoski, Volume preserving mean curvature flows near strictly stable sets in flat torus, J. Differential Equations 276 (2021), 149–186. 10.1016/j.jde.2020.12.010Search in Google Scholar

[24] L. A. Tarshis, J. L. Walker and M. F. X. Gigliotti, Solidification, Ann. Rev. Mater. Sci. 2 (1972), 181–216. 10.1146/annurev.ms.02.080172.001145Search in Google Scholar

[25] C. Wagner, Theorie der Alterung von Niederschlägen durch Umlösen(Ostwald-Reifung), Z. Elektrochem. Berichte Bunsengesell. Phys. Chem. 65 (1961), no. 7–8, 581–591. 10.1002/bbpc.19610650704Search in Google Scholar