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

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Abstract

We consider the (degenerate) parabolic equationu t =G(▽▽u + ug, t) on then-sphereS n. 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 equationu t =G(Δu +cu, ¦x¦,t) on then-ballB n, wherec ≤ λ2(B n).

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Authors and Affiliations

  1. School of Mathematics, University of Minnesota, 206 Church St., S.E., 55455, Minneapolis, MN, USA

    Bennett Chow & Robert Gulliver

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  1. Bennett Chow
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  2. Robert Gulliver
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Chow, B., Gulliver, R. Aleksandrov reflection and nonlinear evolution equations, I: The n-sphere and n-ball. Calc. Var 4, 249–264 (1996). https://doi.org/10.1007/BF01254346

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