Aleksandrov reflection and nonlinear evolution equations, I: The n-sphere and n-ball

  • Bennett Chow
  • Robert Gulliver
Article

Abstract

We consider the (degenerate) parabolic equationut=G(▽▽u + ug, t) on then-sphereSn. This corresponds to the evolution of a hypersurface in Euclidean space by a general function of the principal curvatures, whereu is the support function. Using a version of the Aleksandrov reflection method, we prove the uniform gradient estimate ¦▽u(·,t)¦ <C, whereC depends on the initial conditionu(·, 0) but not ont, nor on the nonlinear functionG. We also prove analogous results for the equationut=Gu +cu, ¦x¦,t) on then-ballBn, wherec ≤ λ2(Bn).

Mathematics subject classification

58G11 35K55 53C21 

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Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • Bennett Chow
    • 1
  • Robert Gulliver
    • 1
  1. 1.School of MathematicsUniversity of MinnesotaMinneapolisUSA

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