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

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 ).