The Mullins-Sekerka problem via the method of potentials

TL;DR

This paper proves the well-posedness of the 2D Mullins-Sekerka problem in certain Sobolev spaces using potential theory, marking a first in unbounded geometries with a novel approach involving singular integral operators.

Contribution

It introduces a new potential theory-based method to analyze the Mullins-Sekerka problem in unbounded domains, establishing well-posedness in Sobolev spaces.

Findings

Well-posedness in Sobolev spaces $H^r( eal)$ for $r ext{ in }(3/2,2)$

First such result in unbounded geometry

Use of singular integral operators to formulate the evolution problem

Abstract

It is shown that the two-dimensional Mullins-Sekerka problem is well-posed in all subcritical Sobolev spaces $H^r(\mathbb{R})$ with $r\in(3/2,2).$ This is the first result where this issue is established in an unbounded geometry. The novelty of our approach is the use of potential theory to formulate the model as an evolution problem with nonlinearities expressed by singular integral operators.