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Brakke's Mean Curvature Flow pp 49–70Cite as

A General Existence Theorem for a Brakke Flow in Codimension One

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Abstract

One of the cornerstone results in [7] is the general existence theorem of a Brakke flow. For any 1 ≤ k < n and any initial rectifiable k-varifold with some minor assumption, Brakke gave a proof of a time-global existence of rectifiable Brakke flow starting from the given data. When the initial data is an integral k-varifold, the obtained flow is also integral in the sense defined in Chap. 2.

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Notes

  1. 1.

    This particular choice of mollifier turned out to be important in the error estimates, see Lemma 4.17.

  2. 2.

    Technically speaking, for the limit of the parabolic Ginzburg–Landau equation, the a.e. integrality of the limit varifolds has not been proved.

References

  1. Allard, W.K., Almgren, F.J., Jr.: The structure of stationary one dimensional varifolds with positive density. Invent. Math. 34(2), 83–97 (1976)

    Article  MathSciNet  Google Scholar 

  2. 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)

    Article  MathSciNet  Google Scholar 

  3. Brakke, K.: The Motion of a Surface by Its Mean Curvature. Mathematical Notes, vol. 20. Princeton University Press, Princeton (1978)

    Google Scholar 

  4. Chen, Y.-G., Giga, Y., Goto, S.: Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations. J. Differ. Geom. 33(3), 749–786 (1991)

    Article  MathSciNet  Google Scholar 

  5. Evans, L.C., Spruck, J.: Motion of level sets by mean curvature. I. J. Differ. Geom. 33(3), 635–681 (1991)

    Article  MathSciNet  Google Scholar 

  6. Evans, L.C., Spruck, J.: Motion of level sets by mean curvature. IV. J. Geom. Anal. 5(1), 77–114 (1995)

    Article  MathSciNet  Google Scholar 

  7. Ilmanen, T.: Convergence of the Allen-Cahn equation to Brakke’s motion by mean curvature. J. Differ. Geom. 38(2), 417–461 (1993)

    Article  MathSciNet  Google Scholar 

  8. Ilmanen, T.: Elliptic regularization and partial regularity for motion by mean curvature. Mem. Amer. Math. Soc. 120, 520 (1994)

    MathSciNet  MATH  Google Scholar 

  9. Kim, L., Tonegawa, Y.: On the mean curvature flow of grain boundaries. Ann. Inst. Fourier (Grenoble) 67(1), 43–142 (2017)

    Article  MathSciNet  Google Scholar 

  10. Lin, F.H.: Some dynamical properties of Ginzburg-Landau vortices. Commun. Pure Appl. Math. 49(4), 323–359 (1996)

    Article  MathSciNet  Google Scholar 

  11. Schulze, F., White, B.: A local regularity theorem for mean curvature flow with triple edges. J. Reine Angew. Math. (to appear). arXiv:1605.06592

    Google Scholar 

  12. Tonegawa, Y.: Integrality of varifolds in the singular limit of reaction-diffusion equations. Hiroshima Math. J. 33(3), 323–341 (2003)

    Article  MathSciNet  Google Scholar 

  13. White, B.: Mean curvature flow (Stanford University), lecture note taken by Otis Chodosh. https://web.math.princeton.edu/~ochodosh/MCFnotes.pdf

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Authors and Affiliations

  1. Tokyo Institute of Technology, Tokyo, Japan

    Yoshihiro Tonegawa

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  1. Yoshihiro Tonegawa
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Tonegawa, Y. (2019). A General Existence Theorem for a Brakke Flow in Codimension One. In: Brakke's Mean Curvature Flow. SpringerBriefs in Mathematics. Springer, Singapore. https://doi.org/10.1007/978-981-13-7075-5_4

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