A symmetry breaking phenomenon for anisotropic harmonic maps from a 2D annulus into $\mathbb S^1$
| Autor: | Contreras, Andres, Lamy, Xavier |
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| Rok vydání: | 2023 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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_\delta(u)=\int_{A_\rho} |\nabla u|^2 +\delta \left( (\nabla\cdot u)^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 qualitative properties very different from the isotropic case $\delta=0$. In particular, $0$-homogeneous critical points of degree $d\notin \lbrace 0,1,2\rbrace$ 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 Rivi\`ere. 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. Comment: The 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 |
| Databáze: | arXiv |
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