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

In a two dimensional annulus $A_{\rho }=\{x\in {\mathbb{R}}^2:\rho <\mathrm{\mid }x\mathrm{\mid }<1\}$, $\rho \in (0,1)$, we characterize $0$-homogeneous minimizers, in $H^1(A_{\rho };{\mathbb{S}}^1)$ with respect to their own boundary conditions, of the anisotropic energy \begin{equation*} E_δ(u)=\int_{A_ρ} |\nabla u|^2 +δ\left( (\nabla\cdot u)^2-(\nabla\times u)^2\right) , dx,\quad δ\in (-1,1). \end{equation*} Even for a small anisotropy $0<\mathrm{\mid }\delta \mathrm{\mid }\ll 1$, we exhibit qualitative properties very different from the isotropic case $\delta =0$. In particular, $0$-homogeneous critical points of degree $d\mathrm{\notin }\{0,1,2\}$ are always local minimizers, but in thick annuli ($\rho \ll 1$) they are not minimizers: the $0$-homogeneous symmetry is broken. One corollary is that entire solutions to the anisotropic Ginzburg-Landau system have a far-field behavior very different from the isotropic case studied by Brezis, Merle and Rivière. The tools we use include: ODE and variational arguments; asymptotic expansions, interpolation inequalities and explicit computations involving near-optimizers of these inequalities for proving that $0$-homogeneous critical points are not minimizers in thick annuli.