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Minimizing movements for forced anisotropic curvature flow of droplets

Shokhrukh Yu. Kholmatov

Abstract

We study forced anisotropic curvature flow of droplets on an inhomogeneous horizontal hyperplane. As in [Bellettini, Kholmatov: J. Math. Pures Appl. (2018)] we establish the existence of smooth flow, starting from a regular droplet and satisfying the prescribed anisotropic Young's law, and also the existence of a $1/2$-Hölder continuous in time minimizing movement solution starting from a set of finite perimeter. Furthermore, we investigate various properties of minimizing movements, including comparison principles, uniform boundedness and the consistency with the smooth flow.

Minimizing movements for forced anisotropic curvature flow of droplets

Abstract

We study forced anisotropic curvature flow of droplets on an inhomogeneous horizontal hyperplane. As in [Bellettini, Kholmatov: J. Math. Pures Appl. (2018)] we establish the existence of smooth flow, starting from a regular droplet and satisfying the prescribed anisotropic Young's law, and also the existence of a -Hölder continuous in time minimizing movement solution starting from a set of finite perimeter. Furthermore, we investigate various properties of minimizing movements, including comparison principles, uniform boundedness and the consistency with the smooth flow.
Paper Structure (24 sections, 25 theorems, 257 equations, 2 figures)

This paper contains 24 sections, 25 theorems, 257 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Existence of GMM
  3. Proof of Theorem \ref{['teo:intro_existGMM']}
  4. Existence of minimizers
  5. Density estimates for minimizers
  6. Flat flows
  7. Uniform boundedness of GMM
  8. Some comparison principles
  9. Discrete comparison and comparison of GMMs
  10. Smooth inner and outer barriers
  11. Comparison of flat flows with truncated Wulff shapes
  12. Smooth $\Phi$-curvature flow of hypersurfaces with boundary
  13. Solvability of \ref{['main_mean_curvature_PDE']}
  14. Hölder spaces
  15. Introducing the parametrization
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $\Phi$ be an elliptic $C^{3+\alpha}$-anisotropy in $\mathbb{R}^n,$$f\in C^{\frac{\alpha}{2},\alpha}(\mathbb{R}_0^+\times \overline{\Omega}),$$\beta\in C^{1+\alpha}(\partial\Omega)$ with $\|\beta\|_\infty<\Phi(\mathbf{e}_n)$ and $E_0\subset\Omega$ be a bounded set such that $\Gamma_0:=\Omega \cap where $\alpha\in (0,1].$ Then there exist a maximal time $T^\dag>0$ and a unique $\Phi$-curvature f

Figures (2)

  • Figure 1: Comparison with Winterbottom shapes.
  • Figure 2: Winterbottom shapes contained in $F_0$ and $F_{\tau*}$.

Theorems & Definitions (43)

  • Theorem 1.1
  • Definition 1.2: Generalized minimizing movements DeGorgi:93
  • Theorem 1.3: Existence of generalized minimizing movements
  • Theorem 1.4: Comparison of GMMs
  • Theorem 1.5
  • Proposition 2.1
  • Theorem 2.2
  • proof
  • Corollary 2.3
  • Corollary 2.4
  • ...and 33 more