Aleksandrov reflection and nonlinear evolution equations, I: The n-sphere and n-ball
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=G(Δu +cu, ¦x¦,t) on then-ballBn, wherec ≤ λ2(Bn).
Mathematics subject classification58G11 35K55 53C21
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