The Existence of a Weak Solution to Volume Preserving Mean Curvature Flow in Higher Dimensions

1. Almgren, F., Taylor, J.E., Wang, L.: Curvature-driven flows: a variational approach.

SIAM J. Control. Optim. 31, 387–438, 1993

2. Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. The Clarendon Press, Oxford University Press, New York (2000)

52 Page 52 of 53

Arch. Rational Mech. Anal. (2023) 247:52

3. Antonopoulou, D., Karali, G., Sigal, I.M.: Stability of spheres under volumepreserving mean curvature flow. Dyn. Partial Differ. Equ. 7, 327–344, 2010

4. Athanassenas, M.: Volume-preserving mean curvature flow of rotationally symmetric

surfaces. Comment. Math. Helv. 72, 52–66, 1997

5. Brakke, K.A.: The Motion of a Surface by Its Mean Curvature. Princeton University

Press, Princeton (1978)

6. Brassel, M., Bretin, E.: A modified phase field approximation for mean curvature

flow with conservation of the volume. Math. Methods Appl. Sci. 34, 1157–1180, 2011

7. Bronsard, L., Stoth, B.: Volume-preserving mean curvature flow as a limit of a

nonlocal Ginzburg-Landau equation. SIAM J. Math. Anal. 28, 769–807, 1997

8. Chen, X., Hilhorst, D., Logak, E.: Mass conserving Allen-Cahn equation and volume

preserving mean curvature flow. Interfaces Free Bound 12, 527–549, 2010

9. Delfour, M.C., Zolésio, J.P.: Shape analysis via oriented distance functions. J. Funct.

Anal. 123, 129–201, 1994

10. Escher, J., Simonett, G.: The volume preserving mean curvature flow near spheres.

Proc. Am. Math. Soc. 126, 2789–2796, 1998

11. Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. CRC

Press, Boca Raton (2015)

12. Fischer, J., Hensel, S., Laux, T., Simon, T.: The local structure of the energy landscape

in multiphase mean curvature flow: Weak-strong uniqueness and stability of evolutions,

arXiv preprint arXiv:2003.05478, 2020

13. Fusco, N.: The classical isoperimetric theorem. Rend. Accad. Sci. Fis. Mat. Napoli. 71,

63–107, 2004

14. Fusco, N., Maggi, F., Pratelli, A.: The sharp quantitative isoperimetric inequality.

Ann. Math. 168, 941–980, 2008

15. Gage, M., On an area-preserving evolution equation for plane curves, Nonlinear problems in geometry (Mobile, Ala.,: Contemp. Math., 51 Amer. Math.

Soc. Providence, R I1986, 51–62, 1985

16. Giusti, E.: Minimal Surfaces and Functions of Bounded Variation. Birkhäuser, Boston

(1984)

17. Golovaty, D.: The volume-preserving motion by mean curvature as an asymptotic

limit of reaction-diffusion equations. Quart. Appl. Math. 55, 243–298, 1997

18. Hensel, S., Laux, T.: A new varifold solution concept for mean curvature flow:

convergence of the Allen-Cahn equation and weak-strong uniqueness, arXiv preprint

arXiv:2109.04233, 2021

19. Huisken, G.: The volume preserving mean curvature flow. J. Reine Angew. Math. 382,

35–48, 1987

20. Huisken, G.: Asymptotic behavior for singularities of the mean curvature flow. J. Differ.

Geom. 31, 285–299, 1990

21. Hutchinson, J.E.: Second fundamental form for varifolds and the existence of surfaces

minimising curvature. Indiana Univ. Math. J. 35, 45–71, 1986

22. Hutchinson, J.E., Tonegawa, Y.: Convergence of phase interfaces in the van der

Waals-Cahn-Hilliard theory. Calc. Var. Partial Differ. Equ. 10, 49–84, 2000

23. Ilmanen, T.: Convergence of the Allen-Cahn equation to Brakke’s motion by mean

curvature. J. Differ. Geom. 38, 417–461, 1993

24. Kim, I., Kwon, D.: Volume preserving mean curvature flow for star-shaped sets. Calc.

Var. Partial Differ. Equ., 2020. https://doi.org/10.1007/s00526-020-01738-0

25. Laux, T.: Weak-strong uniqueness for volume-preserving mean curvature flow, arXiv

preprint arXiv:2205.13040, 2022

26. Laux, T., Otto, F.: Convergence of the thresholding scheme for multi-phase meancurvature flow. Calc. Var. Partial Differ. Equ., 2016. https://doi.org/10.1007/s00526016-1053-0

27. Laux, T., Simon, T.M.: Convergence of the Allen-Cahn equation to multiphase mean

curvature flow. Comm. Pure Appl. Math. 71, 1597–1647, 2018

Arch. Rational Mech. Anal. (2023) 247:52

Page 53 of 53 52

28. Laux, T., Swartz, D.: Convergence of thresholding schemes incorporating bulk effects.

Interfaces Free Bound. 19, 273–304, 2017

29. Li, H.: The volume-preserving mean curvature flow in Euclidean space. Pac. J. Math.

243, 331–355, 2009

30. Liu, C., Sato, N., Tonegawa, Y.: On the existence of mean curvature flow with transport

term. Interfaces Free Bound 12, 251–277, 2010

31. Luckhaus, S., Sturzenhecker, T.: Implicit time discretization for the mean curvature

flow equation. Calc. Var. Partial Differ. Equ. 3, 253–271, 1995

32. Mizuno, M., Tonegawa, Y.: Convergence of the Allen-Cahn equation with Neumann

boundary conditions. SIAM J. Math. Anal. 47, 1906–1932, 2015

33. Modica, L., Mortola, S.: Il limite nella -convergenza di una famiglia di funzionali

ellittici. Boll. Un. Mat. Ital. A 14, 526–529, 1977

34. Mugnai, L., Röger, M.: The Allen-Cahn action functional in higher dimensions. Interfaces Free Bound 10, 45–78, 2008

35. Mugnai, L., Seis, C., Spadaro, E.: Global solutions to the volume-preserving meancurvature flow. Calc. Var. Partial Differ. Equ., 2016. https://doi.org/10.1007/s00526015-0943-x

36. Pisante, A., Punzo, F.: Allen-Cahn approximation of mean curvature flow in Riemannian manifolds, II: Brakke’s flows. Commun. Contemp. Math. 17, 1450041, 2015

37. Pisante, A., Punzo, F.: Allen-Cahn approximation of mean curvature flow in Riemannian manifolds I, uniform estimates. Ann. Sc. Norm. Super. Pisa Cl. Sci. 15, 309–341,

2016

38. Röger, M., Schätzle, R.: On a modified conjecture of De Giorgi. Math. Z. 254, 675–

714, 2006

39. Rubinstein, J., Sternberg, P.: Nonlocal reaction-diffusion equations and nucleation.

IMA J. Appl. Math. 48, 249–264, 1992

40. Simon, L.: Lectures on geometric measure theory, Proc. Centre Math. Anal. Austral.

Nat. Univ. 3, 1983

41. Stuvard, S., Tonegawa, Y.: On the existence of canonical multi-phase Brakke flows,

arXiv preprint arXiv:2109.14415, 2021

42. Takasao, K.: Convergence of the Allen-Cahn equation with constraint to Brakke’s mean

curvature flow. Adv. Differ. Equ. 22, 765–792, 2017

43. Takasao, K.: Existence of weak solution for volume preserving mean curvature flow

via phase field method. Indiana Univ. Math. J. 66, 2015–2035, 2017

44. Takasao, K.: On obstacle problem for Brakke’s mean curvature flow. SIAM J. Math.

Anal. 53, 6355–6369, 2021

45. Takasao, K., Tonegawa, Y.: Existence and regularity of mean curvature flow with

transport term in higher dimensions. Math. Ann. 364, 857–935, 2016

46. Talenti, G.: The standard isoperimetric theorem, Handbook of convex geometry, vol.

A, pp. 73–123. North-Holland, Amsterdam (1993)

47. Tonegawa, Y.: Integrality of varifolds in the singular limit of reaction-diffusion equations. Hiroshima Math. J. 33, 323–341, 2003

48. Tonegawa, Y.: Brakke’s Mean Curvature Flow. SpringerBriefs in Mathematics.

Springer, Singapore (2019)

Keisuke Takasao

Department of Mathematics,

Kyoto University,

Kitashirakawa-Oiwakecho, Sakyo,

Kyoto

606-8502 Japan.

e-mail: k.takasao@math.kyoto-u.ac.jp

(Received January 31, 2022 / Accepted March 28, 2023)

Published online May 15, 2023

© The Author(s) (2023)

...