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Stability for the Surface Diffusion Flow

Antonia Diana, Nicola Fusco, Carlo Mantegazza

TL;DR

This work establishes a dimensionally universal stability result for the surface diffusion flow of hypersurfaces that bound smooth sets in a flat torus. If an initial set is sufficiently close, in a uniform $W^{1,p}$ sense and with small volume symmetric difference, to a strictly stable critical set for the volume-constrained Area functional, then the boundary evolves for all time and converges exponentially to a translate of that critical set. The authors develop a robust high-order energy framework, introduce organizing polynomials ${\mathfrak{p}}_{s}$ and ${\mathfrak{q}}^{s}$ to manage nonlinear terms, and derive uniform interpolation and Calderón–Zygmund estimates to control geometric quantities along the flow. The main result is first demonstrated in dimension $n=4$ with a detailed argument and then extended to general dimensions $n\ge3$, providing a comprehensive stability theory for surface diffusion flow in periodic settings and highlighting the role of translations in the asymptotic limit.

Abstract

We study the global existence and stability of surface diffusion flow (the normal velocity is given by the Laplacian of the mean curvature) of smooth boundaries of subsets of the $n$--dimensional flat torus. More precisely, we show that if a smooth set is ``close enough'' to a strictly stable critical set for the Area functional under a volume constraint, then the surface diffusion flow of its boundary hypersurface exists for all time and asymptotically converges to the boundary of a ``translated'' of the critical set. This result was obtained in dimension $n=3$ by Acerbi, Fusco, Julin and Morini (extending previous results for spheres of Escher, Mayer and Simonett and Elliott and Garcke in dimension $n=2$). Our work generalizes such conclusion to any dimension $n\in\mathbb N$. For sake of clarity, we show all the details in dimension $n=4$ and we list the necessary modifications to the quantities involved in the proof in the general $n$--dimensional case, in the last section.

Stability for the Surface Diffusion Flow

TL;DR

This work establishes a dimensionally universal stability result for the surface diffusion flow of hypersurfaces that bound smooth sets in a flat torus. If an initial set is sufficiently close, in a uniform sense and with small volume symmetric difference, to a strictly stable critical set for the volume-constrained Area functional, then the boundary evolves for all time and converges exponentially to a translate of that critical set. The authors develop a robust high-order energy framework, introduce organizing polynomials and to manage nonlinear terms, and derive uniform interpolation and Calderón–Zygmund estimates to control geometric quantities along the flow. The main result is first demonstrated in dimension with a detailed argument and then extended to general dimensions , providing a comprehensive stability theory for surface diffusion flow in periodic settings and highlighting the role of translations in the asymptotic limit.

Abstract

We study the global existence and stability of surface diffusion flow (the normal velocity is given by the Laplacian of the mean curvature) of smooth boundaries of subsets of the --dimensional flat torus. More precisely, we show that if a smooth set is ``close enough'' to a strictly stable critical set for the Area functional under a volume constraint, then the surface diffusion flow of its boundary hypersurface exists for all time and asymptotically converges to the boundary of a ``translated'' of the critical set. This result was obtained in dimension by Acerbi, Fusco, Julin and Morini (extending previous results for spheres of Escher, Mayer and Simonett and Elliott and Garcke in dimension ). Our work generalizes such conclusion to any dimension . For sake of clarity, we show all the details in dimension and we list the necessary modifications to the quantities involved in the proof in the general --dimensional case, in the last section.
Paper Structure (6 sections, 11 theorems, 189 equations, 1 figure)

This paper contains 6 sections, 11 theorems, 189 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The surface diffusion flow and strictly stable critical sets
  4. Evolution of geometric quantities and basic estimates
  5. Global existence and stability
  6. General dimensions

Key Result

Theorem 1.1

Let $\varphi_0:M\to\mathbb{R}^n$ be a smooth and compact, immersed hypersurface. Then, there exists a unique smooth $\varphi:[0,T)\times M\to\mathbb{R}^n$ such that $\varphi_t=\varphi(t,\cdot)$ is the surface diffusion flow of $\varphi_0$, that is a solution of equation sdf2, for some maximal time o

Figures (1)

  • Figure 1: Periodic critical set classified in Ros: balls, $2$--tori, gyroids and lamellae (from left to right).

Theorems & Definitions (34)

  • Theorem 1.1
  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 3.1
  • Remark 3.2
  • Proposition 3.3
  • Definition 3.4
  • Definition 3.5
  • Lemma 4.1
  • ...and 24 more