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| Format: | Preprint |
| Veröffentlicht: |
2023
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| Online-Zugang: | https://arxiv.org/abs/2311.15758 |
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| _version_ | 1866912018417582080 |
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| author | Contreras, Andres Lamy, Xavier |
| author_facet | Contreras, Andres Lamy, Xavier |
| contents | In a two dimensional annulus $A_ρ=\{x\in \mathbb R^2: ρ<|x|<1\}$, $ρ\in (0,1)$, we characterize $0$-homogeneous minimizers, in $H^1(A_ρ;\mathbb S^1)$ 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 $0<|δ|\ll 1$, we exhibit qualitative properties very different from the isotropic case $δ=0$. In particular, $0$-homogeneous critical points of degree $d\notin \lbrace 0,1,2\rbrace$ are always local minimizers, but in thick annuli ($ρ\ll 1$) 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_15758 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A symmetry breaking phenomenon for anisotropic harmonic maps from a 2D annulus into $\mathbb S^1$ Contreras, Andres Lamy, Xavier Analysis of PDEs In a two dimensional annulus $A_ρ=\{x\in \mathbb R^2: ρ<|x|<1\}$, $ρ\in (0,1)$, we characterize $0$-homogeneous minimizers, in $H^1(A_ρ;\mathbb S^1)$ 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 $0<|δ|\ll 1$, we exhibit qualitative properties very different from the isotropic case $δ=0$. In particular, $0$-homogeneous critical points of degree $d\notin \lbrace 0,1,2\rbrace$ are always local minimizers, but in thick annuli ($ρ\ll 1$) 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. |
| title | A symmetry breaking phenomenon for anisotropic harmonic maps from a 2D annulus into $\mathbb S^1$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2311.15758 |