Translating solutions for Gaußcurvature flows with Neumann boundary conditions

Abstract

We consider strictly convex hypersurfaces which are evolving by the non-parametric logarithmic Gauß curvature flow subject to a Neumann boundary condition. Solutions are shown to converge smoothly to hypersurfaces moving by translation. In particular, for bounded domains we prove that convex functions with prescribed normal derivative satisfy a uniform oscillation estimate.

Capillary-type boundary value problems of mean curvature flow with force and transport terms on a bounded domain

“…Moreover, we get the following result on the large time behavior of solutions of (1.1) by following the argument in [28,34,40].…”

Optimal mass transport for higher dimensional adaptive grid generation

“…The existence, uniqueness, and convergence of globally smooth convex solutions for a general form of the parabolic Monge-Ampère Eq. (23) to the solution of (6) have been studied in [18][19][20] under suitable regularity and structure conditions on X c , X, and the density functions q 0 and q 1 . These results have been extended to include the case where the domains X c and X are squares or cubes in the grid applications [16,17].…”

Gradient estimates for Neumann boundary value problem of Monge–Ampère type equations

“…Note that for convex functions satisfying the Neumann boundary conditions, the gradient estimate can be obtained using the convexity of the function u, see [14,19]. For the Monge-Ampère type equation (1.1), if the matrix A is non-negative definite, then the elliptic solution u is convex and the gradient estimate follows from [14,19].…”

“…For the Monge-Ampère type equation (1.1), if the matrix A is non-negative definite, then the elliptic solution u is convex and the gradient estimate follows from [14,19]. While if the matrix A + KI is non-negative definite for some positive constant K, then the elliptic solution u is semi-convex and the corresponding gradient estimate follows from the Remark at the end of [14,Sec.…”