Hölder gradient estimates for a class of singular or degenerate parabolic equations

Abstract We prove interior Hölder estimates for the spatial gradients of the viscosity solutions to the singular or degenerate parabolic equation u t = | ∇ ⁡ u | κ ⁢ div ⁡ ( | ∇ ⁡ u | p - 2 ⁢ ∇ ⁡ u ) , u_{t}=\lvert\nabla u\rvert^{\kappa}\operatorname{div}(\lvert\nabla u\rvert^{p-% 2}\nabla u), where p ∈ ( 1 , ∞ ) {p\in(1,\infty)} and κ ∈ ( 1 - p , ∞ ) {\kappa\in(1-p,\infty)} . This includes the from L ∞ {L^{\infty}} to C 1 , α {C^{1,\alpha}} regularity for parabolic p-Laplacian equations in both divergence form with κ = 0 {\kappa=0} , and non-divergence form with κ = 2 - p {\kappa=2-p} .