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A priori $L^\infty-$bound for Ginzburg-Landau energy minimizers with divergence penalization

Lia Bronsard, Andrew Colinet, Dominik Stantejsky

Abstract

We consider minimizers $u_\varepsilon$ of the Ginzburg-Landau energy with quadratic divergence penalization on a simply-connected two-dimensional domain $Ω$. On the boundary, strong tangential anchoring is imposed. We prove that minimizers satisfy a $L^\infty$-bound uniform in $\varepsilon$ when $Ω$ has $C^{2,1}-$boundary and that the Lipschitz constant blows up like $\varepsilon^{-1}$ when $Ω$ has $C^{3,1}-$boundary. Our theorem extends to $W^{2,p}-$regularity result for our elliptic system with mixed Dirichlet-Neumann boundary condition.

A priori $L^\infty-$bound for Ginzburg-Landau energy minimizers with divergence penalization

Abstract

We consider minimizers of the Ginzburg-Landau energy with quadratic divergence penalization on a simply-connected two-dimensional domain . On the boundary, strong tangential anchoring is imposed. We prove that minimizers satisfy a -bound uniform in when has boundary and that the Lipschitz constant blows up like when has boundary. Our theorem extends to regularity result for our elliptic system with mixed Dirichlet-Neumann boundary condition.
Paper Structure (11 sections, 9 theorems, 135 equations, 3 figures)

This paper contains 11 sections, 9 theorems, 135 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Outline of the paper:
  3. Preliminaries
  4. Notation
  5. Coordinates
  6. PDE Gluing
  7. Uniform Estimate
  8. Rescaling Argument
  9. Calculations
  10. Polar example
  11. Acknowledgements.

Key Result

Theorem 1.1

Let $\Omega\subset\mathbb{R}^2$ be open, bounded, and simply-connected with $C^{2,1}-$boundary. Then there exist constants $C_1,\varepsilon_{0}>0$, only depending on $k$ and $\Omega$ such that for all $\varepsilon\in (0,\varepsilon_{0})$ and any minimizers $u_\varepsilon$ of def:IntroEnergy subject If $\Omega$ has $C^{3,1}$-boundary then there also exists a constant $C_{2}>0$, only depending on $

Figures (3)

  • Figure 1: Adaptation of the example in Polya30 for dimension $n=2$. For given $k\in\mathbb{R}$, the vector field $u(x,y)=(\frac{\alpha}{2}(x^2+y^2)-\beta, -\gamma xy)$ solves the equation $-\Delta u - k\nabla\mathrm{div} u = 0$ for arbitrary $\beta,\gamma\in\mathbb{R}$ and $\alpha=\frac{k}{k+2}\gamma$ (if $k\neq -2$). If $k=-2$, then set $\gamma=0$ and $\alpha,\beta\in\mathbb{R}$ arbitrary. The above plot depicts the vector field $u(x,y)$ for $k=\beta=\gamma=1$ and $\alpha=\frac{1}{3}$, the color indicates the norm $|u|$. One can see that if $u$ is restricted to a domain $\Omega$ chosen to be a small enough disk around $0$, then the maximum of $|u|$ occurs in the interior of the disk.
  • Figure 2: Illustration of the construction of $X$ and the extension $\widetilde{X}$.
  • Figure 3: We use the unit normal $\nu$ and tangent $\tau$ to parameterize the interior and exterior of the domain $\Omega$ via charts $\widetilde{\psi}_{j}$, in which the reflection $R$ can be represented by a simple sign change of $y_2$.

Theorems & Definitions (18)

  • Theorem 1.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 5.1
  • ...and 8 more