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Convergence of thresholding energies for anisotropic mean curvature flow on inhomogeneous obstacle

Andrea Chiesa, Karel Svadlenka

TL;DR

The paper studies the convergence of thresholding energies for anisotropic mean-curvature flow with an obstacle on an inhomogeneous substrate. By embedding space- and direction-dependent tensions into a BV-family framework and modifying surface tensions to allow a single convolution kernel, the authors construct approximate energies $E_h$ that $ ext{Γ}$-converge to the sharp-interface energy $E$. They formulate the gradient-flow structure in a Finsler geometric setting, derive a BV-type weak solution notion for the obstacle problem, and prove Γ-convergence of the approximate energies while deferring the convergence of the actual thresholding scheme to a subsequent work. The results provide a rigorous foundation for isotropic-to-anisotropic thresholding methods in the presence of spatial inhomogeneity and topological constraints, with potential implications for multi-phase and substrate-affected interface evolution. The framework and techniques lay groundwork for robust numerical schemes that rely on nonlocal convolution-thresholding steps to approximate anisotropic interfacial dynamics on complex substrates.

Abstract

We extend the analysis by Esedoglu and Otto (2015) of thresholding energies for the celebrated multiphase Bence-Merriman-Osher algorithm for computing mean curvature flow of interfacial networks, to the case of differing space-dependent anisotropies. In particular, we address the special setting of an obstacle problem, where anisotropic particles move on an inhomogeneous substrate. By suitable modification of the surface energies we construct an approximate energy that uses a single convolution kernel and is monotone with respect to the convolution width. This allows us to prove that the approximate energies $Γ$-converge to their sharp interface counterpart.

Convergence of thresholding energies for anisotropic mean curvature flow on inhomogeneous obstacle

TL;DR

The paper studies the convergence of thresholding energies for anisotropic mean-curvature flow with an obstacle on an inhomogeneous substrate. By embedding space- and direction-dependent tensions into a BV-family framework and modifying surface tensions to allow a single convolution kernel, the authors construct approximate energies that -converge to the sharp-interface energy . They formulate the gradient-flow structure in a Finsler geometric setting, derive a BV-type weak solution notion for the obstacle problem, and prove Γ-convergence of the approximate energies while deferring the convergence of the actual thresholding scheme to a subsequent work. The results provide a rigorous foundation for isotropic-to-anisotropic thresholding methods in the presence of spatial inhomogeneity and topological constraints, with potential implications for multi-phase and substrate-affected interface evolution. The framework and techniques lay groundwork for robust numerical schemes that rely on nonlocal convolution-thresholding steps to approximate anisotropic interfacial dynamics on complex substrates.

Abstract

We extend the analysis by Esedoglu and Otto (2015) of thresholding energies for the celebrated multiphase Bence-Merriman-Osher algorithm for computing mean curvature flow of interfacial networks, to the case of differing space-dependent anisotropies. In particular, we address the special setting of an obstacle problem, where anisotropic particles move on an inhomogeneous substrate. By suitable modification of the surface energies we construct an approximate energy that uses a single convolution kernel and is monotone with respect to the convolution width. This allows us to prove that the approximate energies -converge to their sharp interface counterpart.

Paper Structure

This paper contains 12 sections, 8 theorems, 161 equations, 2 figures.

Table of Contents

  1. Introduction and Setting
  2. Derivation of gradient flow equations
  3. Finsler-geometric setting and assumptions
  4. First variation of the energy
  5. Gradient flow derivation
  6. Notion of solution
  7. The thresholding scheme and approximate energies
  8. Assumptions
  9. Modified surface tensions
  10. -convergence of thresholding energies
  11. Main theorem
  12. Auxiliary results and proofs

Key Result

Lemma 2.1

Let condition anisotropy phi hold. For any $\lambda\neq 0$, $x \in \overline{\Omega}$, and $\nu, \nu^* \in \mathbb{R}^d\setminus\{0\}$ the following identities hold: In particular, we have, for every $i=1,\dots,d$,

Figures (2)

  • Figure 1: Setup and notation: a particle on a rigid substrate.
  • Figure 2: Definition of the vectors $\nu_S, \nu_P, \nu_{PS}, b_S, b_P, b_{PS}$ and the contact angles $\theta$ and $\theta_c$.

Theorems & Definitions (24)

  • Lemma 2.1: BellettiniPaolini
  • proof
  • Theorem 2.1
  • Remark 2.1: Admissibility of $\xi$
  • Remark 2.2: Volume preservation
  • proof : Proof of Theorem \ref{['first variation energy']}
  • Theorem 2.2
  • proof : Proof of Theorem \ref{['Thm gradient flow']}
  • Definition 2.1: Motion by volume-preserving anisotropic mean curvature flow with obstacle
  • Remark 2.3
  • ...and 14 more