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

Abstract

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 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 $0<|\delta |\ll 1$, we exhibit qualitative properties very different from the isotropic case $\delta =0$. In particular, $0$-homogeneous critical points of degree $d\notin \{0,1,2\}$ 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.

arXiv commentThe third item in Theorem 1.1 is wrong. Minimality of the homogeneous critical point follows from calculations in an upcoming work by P. Bauman and D. Phillips