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Existence of weak solutions to volume-preserving mean curvature flow with obstacles

Jiwoong Jang

TL;DR

This work proves the global-in-time existence of weak solutions to volume-preserving mean curvature flow with obstacles in all dimensions by a phase-field approach. The authors establish convergence of the Allen–Cahn equation with a spatially structured forcing and a multiplier to an $L^2$-flow, linking the limiting velocity to mean curvature and a Lagrange-multiplier term via $v=h- rac{oldsymbol{}}{oldsymbol{ heta}} u$ on $oldsymbol{ extOmega}ackslasholdsymbol{O}$. Key contributions include barrier constructions near obstacles, uniform $L^2$-estimates for the forcing, a monotonicity framework controlling the discrepancy measure, and a rigorous rectifiability and integrality analysis showing that the diffuse interfaces converge to an integral $(d-1)$-varifold. The paper therefore provides a robust diffuse-interface justification for volume-constrained MCF with obstacles and yields a Brakke-type weak solution compatible with obstacle confinement and volume preservation, with potential applications to cell motility models and related geometric flows.

Abstract

We prove the existence of global-in-time weak solutions to volume-preserving mean curvature flow with in the presence of obstacles by the phase field method in all dimensions. Namely, we prove the convergence of solutions to the Allen-Cahn equation with a multiplier to a weak solution to the flow. The choice of the multiplier is motivated from [Mugnai-Seis-Spadaro '16], [Kim-Kwon '20], and [Takasao '23], which enables us to complete the comparison between the multiplier and the forcing that stops the intrusion into the obstacle. We also prove the vanishing of the discrepancy measure by dealing with the forcing term that is now spatially dependent due to the obstacles.

Existence of weak solutions to volume-preserving mean curvature flow with obstacles

TL;DR

This work proves the global-in-time existence of weak solutions to volume-preserving mean curvature flow with obstacles in all dimensions by a phase-field approach. The authors establish convergence of the Allen–Cahn equation with a spatially structured forcing and a multiplier to an -flow, linking the limiting velocity to mean curvature and a Lagrange-multiplier term via on . Key contributions include barrier constructions near obstacles, uniform -estimates for the forcing, a monotonicity framework controlling the discrepancy measure, and a rigorous rectifiability and integrality analysis showing that the diffuse interfaces converge to an integral -varifold. The paper therefore provides a robust diffuse-interface justification for volume-constrained MCF with obstacles and yields a Brakke-type weak solution compatible with obstacle confinement and volume preservation, with potential applications to cell motility models and related geometric flows.

Abstract

We prove the existence of global-in-time weak solutions to volume-preserving mean curvature flow with in the presence of obstacles by the phase field method in all dimensions. Namely, we prove the convergence of solutions to the Allen-Cahn equation with a multiplier to a weak solution to the flow. The choice of the multiplier is motivated from [Mugnai-Seis-Spadaro '16], [Kim-Kwon '20], and [Takasao '23], which enables us to complete the comparison between the multiplier and the forcing that stops the intrusion into the obstacle. We also prove the vanishing of the discrepancy measure by dealing with the forcing term that is now spatially dependent due to the obstacles.
Paper Structure (17 sections, 23 theorems, 155 equations, 1 figure)

This paper contains 17 sections, 23 theorems, 155 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Settings and goals
  3. Prior work and background
  4. Notations and precise setup of phase field
  5. Basic definitions and main results
  6. Settings
  7. Obstacles and well-prepared initial data
  8. Energy dissipation
  9. Barrier functions for obstacles
  10. $L^2$-estimate of the forcing
  11. Rectifiability and integrality of $\mu_t$
  12. Monotonicity formula and upper bound of $\xi^{\varepsilon}$
  13. Upper bound of density ratio
  14. Convergence of $\mu^{\varepsilon}$
  15. Vanishing of $|\xi|$
  16. ...and 2 more sections

Key Result

Theorem 1

Suppose that $d\geq2$ and an open set $U_0\subset\Omega$ has a $C^1$ boundary $M_0$ such that $\overline{O_+}\subset U_0$, $\overline{O_-}\subset \Omega\setminus\overline{U_0}$. Suppose that $O:=O_+\cup O_-$ has $C^{2,\beta}$ boundary for some $\beta\in(0,1)$. Let $\phi^{\varepsilon}$ be the solutio

Figures (1)

  • Figure 1: The forcing term $g^{\varepsilon}$ near $\partial O_+$.

Theorems & Definitions (37)

  • Definition 1: $L^2$-flow MR08
  • Theorem 1
  • Remark 1
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • Lemma 1
  • proof
  • ...and 27 more