Consistency for the surface diffusion flat flow in three dimensions

TL;DR

This paper proves the convergence of a discrete scheme to the unique smooth solution of the surface diffusion equation in three dimensions, under regular initial conditions, advancing understanding of surface evolution modeling.

Contribution

It establishes the convergence of the discrete minimizing movements scheme for surface diffusion in three dimensions, a significant step in mathematical analysis of surface evolution.

Findings

The scheme converges to the unique smooth solution in 3D.

Convergence requires sufficiently regular initial sets.

Provides rigorous proof of scheme's validity for surface diffusion.

Abstract

We investigate the flat flow solution for the surface diffusion equation via the discrete minimizing movements scheme proposed by Cahn and Taylor. We prove that in dimension three the scheme converges to the unique smooth solution of the equation, provided that the initial set is sufficiently regular.