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

TL;DR

This paper establishes uniform bounds and compactness results for anisotropic Ginzburg-Landau minimizers with tangential boundary conditions, revealing defect structures and ruling out boundary vortices in certain cases.

Contribution

It provides new a priori estimates and compactness results for anisotropic Ginzburg-Landau minimizers with tangential anchoring, including defect characterization and vortex exclusion.

Findings

Uniform $L^ abla$ bounds independent of $\\varepsilon$

Convergence to $\\mathbb{S}^1$-valued maps with defects

No boundary vortices in divergence penalized case

Abstract

We consider minimizers $u_\varepsilon$ of the Ginzburg-Landau energy with quadratic divergence or curl penalization on a simply-connected two-dimensional domain $\Omega$. 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.