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

Abstract

We consider minimizers $u_{\epsilon }$ 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 $u_{\epsilon }$ in $L^{\infty }$ uniform in $\epsilon$ and that the Lipschitz constant of $u_{\epsilon }$ blows up like ${\epsilon }^{-1}$. 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.

37 pages, 3 figures