Advertisement

Definition of the Brakke Flow

  • Yoshihiro Tonegawa
Chapter
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)

Abstract

Suppose that we have a family of smooth surfaces {Γ(t)}t ∈ [0,T) for some T ≤ with no boundaries in \(U\subset \mathbb {R}^n\). Let v = v(Γ(t), x) be the normal velocity vector of Γ(t) at x ∈ Γ(t). Here we consider how one may characterize the normal velocity using integration. The reason for such a pursuit is that, in the end, we want to replace Γ(t) by a general varifold. To do so, let \(\phi \in C_c^1(U\times [0,T);\mathbb {R}^+)\) be a non-negative “test function”.

References

  1. 6.
    Bethuel, F., Orlandi, G., Smets, D.: Convergence of the parabolic Ginzburg-Landau equation to motion by mean curvature. Ann. Math. (2) 163(1), 37–163 (2006)MathSciNetCrossRefGoogle Scholar
  2. 7.
    Brakke, K.: The Motion of a Surface by Its Mean Curvature. Mathematical Notes, vol. 20. Princeton University Press, Princeton (1978)Google Scholar
  3. 18.
    Ilmanen, T.: Convergence of the Allen-Cahn equation to Brakke’s motion by mean curvature. J. Differ. Geom. 38(2), 417–461 (1993)MathSciNetCrossRefGoogle Scholar
  4. 19.
    Ilmanen, T.: Elliptic regularization and partial regularity for motion by mean curvature. Mem. Amer. Math. Soc. 120, 520 (1994)MathSciNetzbMATHGoogle Scholar
  5. 21.
    Kasai, K., Tonegawa, Y.: A general regularity theory for weak mean curvature flow. Calc. Var. PDE. 50, 1–68 (2014)MathSciNetCrossRefGoogle Scholar
  6. 22.
    Kim, L., Tonegawa, Y.: On the mean curvature flow of grain boundaries. Ann. Inst. Fourier (Grenoble) 67(1), 43–142 (2017)MathSciNetCrossRefGoogle Scholar
  7. 26.
    Lin, F.H.: Some dynamical properties of Ginzburg-Landau vortices. Commun. Pure Appl. Math. 49(4), 323–359 (1996)MathSciNetCrossRefGoogle Scholar
  8. 30.
    Metzger, J., Schulze, F.: No mass drop for mean curvature flow of mean convex hypersurfaces. Duke Math. J. 142(2), 283–312 (2008)MathSciNetCrossRefGoogle Scholar
  9. 35.
    Takasao, K., Tonegawa, Y.: Existence and regularity of mean curvature flow with transport term in higher dimensions. Math. Ann. 364(3–4), 857–935 (2016)MathSciNetCrossRefGoogle Scholar
  10. 37.
    Tonegawa, Y.: A second derivative Hölder estimate for weak mean curvature flow. Adv. Calc. Var. 7(1), 91–138 (2014)MathSciNetCrossRefGoogle Scholar
  11. 40.
    White, B.: The size of the singular set in mean curvature flow of mean-convex surfaces. J. Am. Math. Soc. 13(3), 665–695 (2000)CrossRefGoogle Scholar
  12. 41.
    White, B.: The nature of singularities in mean curvature flow of mean-convex surfaces. J. Am. Math. Soc. 16(1), 123–138 (2003)CrossRefGoogle Scholar

Copyright information

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  • Yoshihiro Tonegawa
    • 1
  1. 1.Tokyo Institute of TechnologyTokyoJapan

Personalised recommendations