We consider the (degenerate) parabolic equation ut = 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 ut = G(Δu + cu, |x|, t) on the n-ball Bn, where c ≤ λ2(Bn).
|Original language||English (US)|
|Number of pages||16|
|Journal||Calculus of Variations and Partial Differential Equations|
|State||Published - Apr 1996|