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Varifold solutions to Volume-Preserving Mean Curvature Flow: existence and weak-strong uniqueness

Andrea Poiatti

Abstract

In this contribution we introduce a novel weak solution concept for two-phase volume-preserving mean curvature flow, having both properties of unconditional global-in-time existence and weak-strong uniqueness. These solutions extend the ones proposed by Hensel-Laux [J. Differential Geom. 130, 209-268 (2025)] for the standard mean curvature flow, and consist in evolving varifolds coupled with the phase volumes by a transport equation. First, we show that, in the same setting as in Takasao [Arch. Ration. Mech. Anal. 247, 52 (2023)], any sharp interface limit of solutions to a slightly modified nonlocal Allen-Cahn equation is a varifold solution according to our new definition. Secondly, we crucially introduce a new notion of volume-preserving gradient-flow calibrations, allowing the extended velocity vector field to point in the normal direction on the interface. We show that any sufficiently regular strong solution is calibrated in this sense. Finally, we prove that any classical solution to volume-preserving mean curvature flow, which is then automatically a calibrated flow, is unique in the class of our new varifold solutions.

Varifold solutions to Volume-Preserving Mean Curvature Flow: existence and weak-strong uniqueness

Abstract

In this contribution we introduce a novel weak solution concept for two-phase volume-preserving mean curvature flow, having both properties of unconditional global-in-time existence and weak-strong uniqueness. These solutions extend the ones proposed by Hensel-Laux [J. Differential Geom. 130, 209-268 (2025)] for the standard mean curvature flow, and consist in evolving varifolds coupled with the phase volumes by a transport equation. First, we show that, in the same setting as in Takasao [Arch. Ration. Mech. Anal. 247, 52 (2023)], any sharp interface limit of solutions to a slightly modified nonlocal Allen-Cahn equation is a varifold solution according to our new definition. Secondly, we crucially introduce a new notion of volume-preserving gradient-flow calibrations, allowing the extended velocity vector field to point in the normal direction on the interface. We show that any sufficiently regular strong solution is calibrated in this sense. Finally, we prove that any classical solution to volume-preserving mean curvature flow, which is then automatically a calibrated flow, is unique in the class of our new varifold solutions.

Paper Structure

This paper contains 10 sections, 9 theorems, 178 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results on De Giorgi-type varifold solutions for volume-preserving mean curvature flow
  4. Definition, existence results, and consistency
  5. Gradient-flow calibration and weak-strong uniqueness
  6. Proof of Theorem \ref{['thm:existence']}
  7. Proof of Lemma \ref{['lemma:further']}
  8. Proof of Theorem \ref{['thmcalibration']}. Existence of a gradient-flow calibration for volume preserving mean curvature flow.
  9. Weak-strong uniqueness: Proof of Theorem \ref{['theo:weakStrongUniqueness']}
  10. An alternative proof for weak-strong uniqueness for integral varifolds

Key Result

Theorem 3.5

Let $\Omega=\mathbb T^d$ and let $u_\varepsilon$ denote the solution to the modified Allen--Cahn equation with well-prepared initial conditions $u_{\varepsilon,0}$ in the sense of Definition well-prepared. Then there exist a measurable function $\chi \colon \Omega \times (0,\infty ) \to \{0,1\}$, a Furthermore, the pair $(\chi,\mu)$ is a varifold solution to the volume-preserving mean curvature f

Theorems & Definitions (26)

  • Definition 3.1: De Giorgi-type varifold solutions for volume-preserving mean curvature flow
  • Remark 3.2: Rectifiability of the varifold
  • Remark 3.3: Local time integrability of the mean curvature vector $\mathbf{H}$
  • Remark 3.4
  • Theorem 3.5: Convergence of the modified nonlocal Allen--Cahn equation to a De Giorgi solution
  • Remark 3.6
  • Remark 3.7
  • Definition 3.8: Strong solution to volume-preserving mean curvature flow
  • Lemma 3.9
  • Definition 3.10
  • ...and 16 more