Advertisement

Gradient estimates for mean curvature flow with Neumann boundary conditions

  • Masashi Mizuno
  • Keisuke Takasao
Article
  • 106 Downloads

Abstract

We study the mean curvature flow of graphs both with Neumann boundary conditions and transport terms. We derive boundary gradient estimates for the mean curvature flow. As an application, the existence of the mean curvature flow of graphs is presented. A key argument is a boundary monotonicity formula of a Huisken type derived using reflected backward heat kernels. Furthermore, we provide regularity conditions for the transport terms.

Keywords

Mean curvature flow Boundary gradient estimates Boundary monotonicity formula 

Mathematics Subject Classification

Primary 35K93 Secondary 53C44 35B65 

References

  1. 1.
    Allard, W.K.: On the first variation of a varifold: boundary behavior. Ann. of Math. 101(2), 418–446 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Altschuler, S.J., Wu, L.F.: Translating surfaces of the non-parametric mean curvature flow with prescribed contact angle. Calc. Var. Partial Differ. Equ. 2, 101–111 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Buckland, J.A.: Mean curvature flow with free boundary on smooth hypersurfaces. J. Reine Angew. Math. 586, 71–90 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Colding, T.H., Minicozzi II, W.P.: Sharp estimates for mean curvature flow of graphs. J. Reine Angew. Math. 574, 187–195 (2004)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Ecker, K.: Regularity Theory for Mean Curvature Flow, vol. 57. Birkhäuser Boston Inc, Boston (2004)CrossRefzbMATHGoogle Scholar
  6. 6.
    Ecker, K., Huisken, G.: Interior curvature estimates for hypersurfaces of prescribed mean curvature. Ann. Inst. H. Poincaré Anal. Non Linéaire 6, 251–260 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ecker, K., Huisken, G.: Mean curvature evolution of entire graphs. Ann. Math. 130(2), 453–471 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ecker, K., Huisken, G.: Interior estimates for hypersurfaces moving by mean curvature. Invent. Math. 105, 547–569 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Edelen, N.: Convexity estimates for mean curvature flow with free boundary. Adv. Math. 294, 1–36 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Edelen, N.: The free-boundary Brakke flow, arXiv preprint arXiv:1602.03614 (2016)
  11. 11.
    Grüter, M., Jost, J.: Allard type regularity results for varifolds with free boundaries. Ann. Scuola Norm. Super. Pisa Cl. Sci. 13(4), 129–169 (1986)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Huisken, G.: Nonparametric mean curvature evolution with boundary conditions. J. Differ. Equ. 77, 369–378 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Huisken, G.: Asymptotic behavior for singularities of the mean curvature flow. J. Differ. Geom. 31, 285–299 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Ilmanen, T.: Convergence of the Allen–Cahn equation to Brakke’s motion by mean curvature. J. Differ. Geom. 38, 417–461 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kasai, K., Tonegawa, Y.: A general regularity theory for weak mean curvature flow. Calc. Var. Partial Differ. Equ. 50, 1–68 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Koeller, A.N.: Regularity of mean curvature flows with Neumann free boundary conditions. Calc. Var. Partial Differ. Equ. 43, 265–309 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Ladyženskaja, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and quasilinear equations of parabolic type, Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., (1967)Google Scholar
  18. 18.
    Liu, C., Sato, N., Tonegawa, Y.: Two-phase flow problem coupled with mean curvature flow. Interfaces Free Bound. 14, 185–203 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Liu, C., Walkington, N.J.: An Eulerian description of fluids containing visco-hyperelastic particles. Arch. Ration. Mech. Anal. 159, 229–252 (2001)Google Scholar
  20. 20.
    Mizuno, M., Tonegawa, Y.: Convergence of the Allen–Cahn equation with Neumann boundary conditions. SIAM J. Math. Anal. 47, 1906–1932 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Stahl, A.: Convergence of solutions to the mean curvature flow with a Neumann boundary condition. Calc. Var. Partial Differ. Equ. 4, 421–441 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Stahl, A.: Regularity estimates for solutions to the mean curvature flow with a Neumann boundary condition. Calc. Var. Partial Differ. Equ. 4, 385–407 (1996)Google Scholar
  23. 23.
    Stone, A.: A boundary regularity theorem for mean curvature flow. J. Differ. Geom. 44, 371–434 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Stone, A.G.: Singular and boundary behaviour in the mean curvature flow of hypersurfaces, Ph.D. thesis, Stanford University (1994)Google Scholar
  25. 25.
    Takasao, K.: Gradient estimates and existence of mean curvature flow with transport term. Differ. Integral Equ. 26, 141–154 (2013)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Tonegawa, Y.: A second derivative Hölder estimate for weak mean curvature flow. Adv. Calc. Var. 7, 91–138 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Wheeler, G., Wheeler, V.-M.: Mean curvature flow with free boundary outside a hypersphere, arXiv preprint arXiv:1405.7774, (2014)
  28. 28.
    Wheeler, V.-M.: Mean curvature flow of entire graphs in a half-space with a free boundary. J. Reine Angew. Math. 690, 115–131 (2014)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Wheeler, V.-M.: Non-parametric radially symmetric mean curvature flow with a free boundary. Math. Z. 276, 281–298 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Ziemer, W.P.: Weakly Differentiable Functions, Graduate Texts in Mathematics, vol. 120. Springer-Verlag, New York (1989)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  1. 1.Department of Mathematics, College of Science and TechnologyNihon UniversityTokyoJapan
  2. 2.Graduate School of Mathematical SciencesUniversity of TokyoTokyoJapan

Personalised recommendations