Optimal relaxation to a planar interface in the Mullins-Sekerka problem

TL;DR

This paper extends a relaxation method to analyze convergence rates of interfaces in the Mullins-Sekerka problem, achieving optimal results by relating distance, energy, and dissipation in higher dimensions.

Contribution

It introduces an extension of a relaxation method from 1D to higher dimensions for Mullins-Sekerka, accounting for interface curvature effects.

Findings

Achieves optimal convergence rates under certain initial conditions.

Extends relaxation method to higher-dimensional interface problems.

Highlights the role of curvature in convergence analysis.

Abstract

We analyze the convergence rates to a planar interface in the Mullins-Sekerka model by applying a relaxation method based on relationships among distance, energy, and dissipation. The relaxation method was developed by two of the authors in the context of the 1-d Cahn-Hilliard equation and the current work represents an extension to a higher dimensional problem in which the curvature of the interface plays an important role. The convergence rates obtained are optimal given the assumptions on the initial data.