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Sharp Convergence to the Half-Space for Mullins-Sekerka in the Plane

Wenhui Shi, Maria G. Westdickenberg, Michael Westdickenberg

TL;DR

This work develops an intrinsic interface-based analysis for the planar Mullins-Sekerka evolution by equipping the interface with a natural distance, energy, and dissipation defined directly on Γ. It builds a geometric-harmonic-analysis framework (δ-flat AR domains, boundary-layer potentials, and Dirichlet-to-Neumann maps) to study the gradient-flow structure, deriving differential and algebraic estimates that lead to a sharp, leading-order convergence rate to the flat interface. The authors prove a main theorem giving universal, small-ε0 dependent bounds that yield convexity of the energy and optimal t−1 and t−2 decay rates for E and D, respectively, matching linearized behavior in the plane. The results advance the understanding of noncompact Mullins-Sekerka evolution by providing intrinsic, sharp convergence criteria and constants in the plane, leveraging curvature control, Allard-type regularity, and singular-integral theory on interfaces.

Abstract

We revisit the HED Method for the Mullins-Sekerka evolution in the plane. We identify a natural notion of distance, intrinsic to the interface itself. Using this distance, the energy, and the dissipation, we develop natural assumptions on the flow and, assuming existence of a solution satisfying these conditions, establish not just the algebraic rate (previously derived by Chugreeva, Otto, and M. G. Westdickenberg) but also the sharp leading order constant for the convergence to the flat limiting interface.

Sharp Convergence to the Half-Space for Mullins-Sekerka in the Plane

TL;DR

This work develops an intrinsic interface-based analysis for the planar Mullins-Sekerka evolution by equipping the interface with a natural distance, energy, and dissipation defined directly on Γ. It builds a geometric-harmonic-analysis framework (δ-flat AR domains, boundary-layer potentials, and Dirichlet-to-Neumann maps) to study the gradient-flow structure, deriving differential and algebraic estimates that lead to a sharp, leading-order convergence rate to the flat interface. The authors prove a main theorem giving universal, small-ε0 dependent bounds that yield convexity of the energy and optimal t−1 and t−2 decay rates for E and D, respectively, matching linearized behavior in the plane. The results advance the understanding of noncompact Mullins-Sekerka evolution by providing intrinsic, sharp convergence criteria and constants in the plane, leveraging curvature control, Allard-type regularity, and singular-integral theory on interfaces.

Abstract

We revisit the HED Method for the Mullins-Sekerka evolution in the plane. We identify a natural notion of distance, intrinsic to the interface itself. Using this distance, the energy, and the dissipation, we develop natural assumptions on the flow and, assuming existence of a solution satisfying these conditions, establish not just the algebraic rate (previously derived by Chugreeva, Otto, and M. G. Westdickenberg) but also the sharp leading order constant for the convergence to the flat limiting interface.

Paper Structure

This paper contains 15 sections, 24 theorems, 288 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Convergence rates and convexity
  3. Model formulation
  4. Geometric Harmonic Analysis
  5. Flatness from Oscillation Control
  6. Minimal Surface Theory
  7. Singular Integral Operators
  8. Function Spaces
  9. Chord-arc Domains in the Plane
  10. Mullins-Sekerka Evolution
  11. Gradient Flow Structure
  12. Static/algebraic estimates
  13. Differential estimates
  14. Optimal Convergence Rates
  15. Proofs

Key Result

Proposition 1.1

Suppose that the energy $E$, defined on some Hilbert space with norm $|\cdot|$, is convex, twice continuously differentiable, and bounded below with minimum energy $0$. Suppose there exists a global minimizer $u_*$ of $E$, so that $E(u_*)=0$. Consider the gradient flow and the related squared distance and dissipation Denote the initial values $H_0:=H(u_0),\;E_0:=E(u_0),\;D_0:=D(u_0)$. The gradie

Figures (1)

  • Figure 1: Spiral point (left: $\epsilon=0.8$, right: $\epsilon=0.4$). Note that this is only a very incomplete depiction of the situation. In reality, the interface spirals infinitely many times around the origin. The limited resolution cannot make this effect visible.

Theorems & Definitions (80)

  • Proposition 1.1
  • proof
  • Remark 1.2
  • Example 1.3
  • Corollary 1.4
  • proof
  • Theorem 1.5: Convergence rates
  • Remark 1.6: Constants
  • Definition 2.1: Ahlfors Regularity
  • Theorem 2.2: First Variation of Perimeter
  • ...and 70 more