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A convergence result for a minimizing movement scheme for mean curvature flow with prescribed contact angle in a curved domain

Tokuhiro Eto, Yoshikazu Giga

Abstract

We consider a minimizing movement scheme of Chambolle type for the mean curvature flow equation with prescribed contact angle condition in a smooth bounded domain in $\mathbb{R}^d$ ($d\geq2$). We prove that an approximate solution constructed by the proposed scheme converges to the level-set mean curvature flow with prescribed contact angle provided that the domain is convex and that the contact angle is away from zero under some control of derivatives of given prescribed angle. We actually prove that an auxiliary function corresponding to the scheme uniformly converges to a unique viscosity solution to the level-set equation with an oblique {derivative} boundary condition corresponding to the prescribed boundary condition.

A convergence result for a minimizing movement scheme for mean curvature flow with prescribed contact angle in a curved domain

Abstract

We consider a minimizing movement scheme of Chambolle type for the mean curvature flow equation with prescribed contact angle condition in a smooth bounded domain in (). We prove that an approximate solution constructed by the proposed scheme converges to the level-set mean curvature flow with prescribed contact angle provided that the domain is convex and that the contact angle is away from zero under some control of derivatives of given prescribed angle. We actually prove that an auxiliary function corresponding to the scheme uniformly converges to a unique viscosity solution to the level-set equation with an oblique {derivative} boundary condition corresponding to the prescribed boundary condition.
Paper Structure (8 sections, 24 theorems, 141 equations, 4 figures)

This paper contains 8 sections, 24 theorems, 141 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Convex analysis
  4. Viscosity solution
  5. Capillary Chambolle type scheme
  6. Convergence of the proposed scheme
  7. Conclusion
  8. Acknowledgements

Key Result

Theorem 1.1

Assume that $\Omega$ is a bounded convex set in $\mathbb{R}^d$ whose boundary is sufficiently regular so that the comparison principle holds for eq:4_LevelSetEquationBdd. Suppose that $\beta\in C^1(\partial\Omega)$ and $\|\beta\|_\infty < 1$. Assume that $|\nabla_{\partial\Omega}\beta(x)|\leq k(x)$

Figures (4)

  • Figure 1: The location of important points associated with $z\in\Omega$.
  • Figure 2: The boundaries of the super level sets.
  • Figure 3: The graphs of a translating soliton and the level set of $\varphi$.
  • Figure 4: Level sets of $s_{\mu-\varepsilon h - \overline{C} h,\overline{\beta}}$, $s_{\mu+\varepsilon h + \underline{C} h, \underline{\beta}}$ and $\varphi$.

Theorems & Definitions (58)

  • Theorem 1.1
  • Corollary 1.1
  • Definition 2.1: Subdifferential
  • Definition 2.2: Conjugate function
  • Proposition 2.1: Fenchel identity
  • proof
  • Remark 2.1: Characterization of conjugate functions
  • Definition 2.3: Viscosity solution
  • Definition 2.4: Degenerate ellipticity
  • Theorem 2.1: Comparison principle
  • ...and 48 more