Time-interior gradient estimates for quasilinear parabolic equations

Abstract

Bounded smooth solutions of the Dirichlet and Neumann problems for a wide variety of quasilinear parabolic equations, including graphical anisotropic mean curvature flows, have gradient bounded in terms of oscillation and elapsed time.

Translating solutions of non-parametric mean curvature flows with capillary-type boundary value problems

“…In (1) and (2), when q = 0, it is the capillary problem and when q = 1, it is the Neumann boundary value problem.…”

“…Guan proved that in both cases the solution asymptotically approaches the solution of the corresponding stationary equation. But in [12], ϕ(u, Du) must have crucial monotonicity with respect to u. Andrews and Clutterbuck [2] studied the mean curvature equation on a convex domain with the homogeneous Neumann boundary and initial data u 0 ∈ C(Ω) for q = 1. Recently, the third author in this note deduced the existence theorem for the mean curvature flow of graphs with general Neumann boundary condition in [33] by applying the technique in [26].…”

Gradient Estimates via Two-Point Functions for Parabolic Equations Under Ricci Flow

“…Andrews and Clutterbuck [2,3] and Andrews [1] studied two-point estimates and their applications in a variety of geometric contexts. Recently, Andrews and Xiong [4] where the left-hand side of (1.1) is continuous on R × T M * x × L 2 s (T M), α and β are nonnegative functions and β(s, t) > 0 for t > 0.…”

Moduli of Continuity for Viscosity Solutions on Manifolds

“…For domains in Euclidean spaces, B. Andrews and J. Clutterbuck [AC09] proved that the modulus of continuity for a regular solution of (1.1) is a viscosity subsolution of the onedimensional equation φ t = α(φ ′ , t)φ ′′ . Recently this was shown by the first author [Li] to be true for viscosity solutions as well.…”