A priori estimates and $η-$compactness for anisotropic Ginzburg-Landau minimizers with tangential anchoring

TL;DR

This paper analyzes a priori estimates and $\eta$-compactness for anisotropic Ginzburg–Landau minimizers with tangential boundary anchoring in two dimensions. The authors establish uniform $L^\infty$ bounds and an $\varepsilon^{-1}$-scaled Lipschitz bound via a boundary-extension framework for the curl-energy, and they prove $\eta$-compactness for both divergence- and curl-penalized energies, enabling a precise description of defect sets. A lower-bound/compactness argument then yields convergence to $\mathbb{S}^1$-valued maps away from a finite vortex set $\Sigma$, with the divergence-penalized problem admitting a single interior vortex and excluding boundary vortices. In the curl-penalized case, the defect set is either a single interior vortex or two boundary half-defects, reflecting the influence of the tangential anchoring and anisotropic elastic terms on vortex localization. Overall, the work provides a rigorous ε→0 description of vortex configurations under tangential boundary conditions and anisotropic elasticity, clarifying when boundary vortices can occur and how they interact with splay/bend penalties.

Abstract

We consider minimizers $u_\varepsilon$ of the Ginzburg-Landau energy with quadratic divergence or curl penalization on a simply-connected two-dimensional domain $Ω$. On the boundary, strong tangential anchoring is imposed. We prove a priori estimates for $u_\varepsilon$ in $L^\infty$ uniform in $\varepsilon$ and that the Lipschitz constant of $u_\varepsilon$ blows up like $\varepsilon^{-1}$. We then deduce compactness for a subsequence that converges to an $\mathbb{S}^1-$valued map with either one interior point defect or two boundary half-defects. We conclude our study with a proof that no boundary vortices can occur in the divergence penalized case.

Figures (3)

• Figure 1: Plot of energy minimal vector fields $u_\varepsilon$ of $E_\varepsilon$ with divergence penalization (left) and curl penalization (right) for $k=1$ and $\varepsilon=0.01$ on the unit disk subject to the boundary condition $\langle u_\varepsilon,x\rangle=0$. The divergence penalized minimizers shows a divergence-free interior degree $1$ vortex, while in the curl-penalized case two opposing half-vortices on the boundary are preferred.
• Figure 2: Plot of a minimizer $u_\varepsilon$ of $E_\varepsilon$ with curl-penalization and $u_\varepsilon(x)=x^\perp$ imposed on the boundary. Note that this is not the energy minimal configuration for $\langle u_\varepsilon,x\rangle=0$ (see Figure \ref{['fig:disk:strAnch:div_curl']}). The Dirichlet condition forces an interior defect of type $\frac{x}{|x|}$ at the origin, see also LKM2006.
• Figure 3: Plot of energy minimal vector fields $u_\varepsilon$ of $E_\varepsilon$ with divergence penalization (left) and curl penalization (right) for $k=1$ and $\varepsilon=0.01$ on a peanut shaped domain subject to the boundary condition $\langle u_\varepsilon,n\rangle=0$. For $\varepsilon$ sufficiently small, the divergence penalized minimizers shows a divergence-free interior degree $1$ vortex. In the curl-penalized case, two half-vortices on the boundary placed as far from each other as possible are optimal.

Theorems & Definitions (29)

• Theorem 1.1
• Proposition 2.1
• proof
• Lemma 2.2: bcs24
• Theorem 2.3
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• proof : Proof of Theorem \ref{['thm:a-priori-curl']}