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In a two dimensional annulus $A_\rho=\{x\in \mathbb R^2: \rho<|x|<1\}$,$\rho\in (0,1)$, we characterize $0$-homogeneous minimizers, in$H^1(A_\rho;\mathbb S^1)$ with respect to their own boundary conditions, of theanisotropic energy \begin{equation*}  E_\delta(u)=\int_{A_\rho} |\nabla u|^2 +\delta \left( (\nabla\cdotu)^2-(\nabla\times u)^2\right) \, dx,\quad \delta\in (-1,1). \end{equation*}Even for a small anisotropy $0<|\delta|\ll 1$, we exhibit qualitativeproperties very different from the isotropic case $\delta=0$. In particular,$0$-homogeneous critical points of degree $d\notin \lbrace 0,1,2\rbrace$ arealways local minimizers, but in thick annuli ($\rho\ll 1$) they are notminimizers: the $0$-homogeneous symmetry is broken. One corollary is thatentire solutions to the anisotropic Ginzburg-Landau system have a far-fieldbehavior very different from the isotropic case studied by Brezis, Merle andRivi\`ere. The tools we use include: ODE and variational arguments; asymptoticexpansions, interpolation inequalities and explicit computations involvingnear-optimizers of these inequalities for proving that $0$-homogeneous criticalpoints are not minimizers in thick annuli.

A symmetry breaking phenomenon for anisotropic harmonic maps from a 2D annulus into $\mathbb S^1$