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manuscripta mathematica

, Volume 158, Issue 1–2, pp 75–84 | Cite as

Mean curvature flow of graphs with Neumann boundary conditions

  • Jinju XuEmail author
Article
  • 48 Downloads

Abstract

In this paper, we study the mean curvature flow of graphs with Neumann boundary conditions. The main aim is to use the maximum principle to get the boundary gradient estimate for solutions. As an application, we obtain the corresponding long time existence for the mean curvature flow of graphs.

Mathematics Subject Classification

Primary 35B45 Secondary 35J92 35B50 

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References

  1. 1.
    Altschuler, S.J., Wu, L.F.: Translating surfaces of the non-parametric mean curvature flow with prescribed contact angle. Calc. Var. 2, 101–111 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ben, A., Julie, C.: Time-interior gradient estimates for quasilinear parabolic equations. Indiana Univ. Math. J. 58(1), 351–380 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ecker, K., Huisken, G.: Interior curvature estimates for hypersurfaces of prescribed mean curvature. Ann. Inst. H. Poincare’ Anal. Non Lin’eaire 6, 251–260 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Guan, B.: Mean curvature motion of non-parametric hypersurfaces with contact angle condition. In: Peters AK (ed) Elliptic and Parabolic Methods in Geometry, pp. 47–56. Wellesley (MA) (1996)Google Scholar
  5. 5.
    Huisken, G.: Non-parametric mean curvature evolution with boundary conditions. J. Differ. Equ. 77, 369–378 (1989)CrossRefzbMATHGoogle Scholar
  6. 6.
    Lieberman, G.: Gradient estimates for capillary-type problems via the maximum principle. Commun. Part. Differ. Equ. 13(1), 33–59 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ma, X.N., Wang, P.H., Wei, W.: Constant mean curvature surfaces and mean curvature flow with non-zero Neumann boundary conditions on strictly convex domains. J. Funct. Anal. 274(1), 252–277 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ma, X.N., Xu, J.J.: Gradient estimates of mean curvature equations with Neumann boundary condition. Adv. Math. 290, 1010–1039 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Simon, L., Spruck, J.: Existence and regularity of a capillary surface with prescribed contact angle. Arch. Rational Mech. Anal. 61, 19–34 (1976)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsShanghai Normal UniversityShanghaiChina

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