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

We consider the (degenerate) parabolic equation u_t = G(∇∇u + ug, t) on the n-sphere Sⁿ. This corresponds to the evolution of a hypersurface in Euclidean space by a general function of the principal curvatures, where u is the support function. Using a version of the Aleksandrov reflection method, we prove the uniform gradient estimate |∇u(·, t)| ≤ C, where C depends on the initial condition u(·, 0) but not on t, nor on the nonlinear function G. We also prove analogous results for the equation u_t = G(Δu + cu, |x|, t) on the n-ball Bⁿ, where c ≤ λ₂(Bⁿ).