Asymptotic behavior for minimizers of an anisotropic Ginzburg-Landau functional

Abstract

Minimizers $u_{\varepsilon}$ of some anisotropic Ginzburg-Landau functional $E_{\varepsilon}$ defined in (1.4) below on a smooth domain ${\Omega}\subset R^2$ with smooth boundary data of degree $d$ are shown to subconverge (as ${\varepsilon}\to 0$) locally in $C^{1+{\alpha}}$ away from finitely many points $a_1,\ldots,a_N$ to an anisotropic harmonic map $u_*$: ${\Omega}\setminus \{a_1,\ldots,a_N\}$ $\to$ $R^2$ where $\{a_1,\ldots,a_N\}\subset{\Omega} \cap a^{-1}(m)$, $N$ is related to the degree $d$, $m,a^{-1}(m)$ are defined in the following.