AI 中文总结
针对受阻铁磁体四阶模型，证明在H>1/4时拓扑类Q=±1中极小元存在且序列预紧，确定H=1/4为谱阈值，并刻画强场下重标度极小元的渐近行为。
AI 中文摘要
我们研究了一个描述二维受阻铁磁体的四阶变分模型，该模型包含竞争性交换相互作用以及强度为$H>0$的外加磁场。对于每个$H>1/4$，我们证明了能量在拓扑类$Q=\pm1$中允许极小元存在，并且每个极小化序列在$H^2$模平移意义下是预紧的。主要困难在于谱强制性在$H\downarrow1/4$时退化。利用亥姆霍兹圆平均恒等式，我们证明了每个非零度构型的剩余能量具有一致正下界，即使在退化端点也如此。结合球值$H^2$-分裂构造，这提供了一个阈值稳定的束缚不等式，并在整个强制性区域中产生了紧性。我们还确定了$H=1/4$为尖锐谱阈值。低于该阈值时，能量从下方无界；而在阈值处，非零度构型保持正的能量势垒，而零度Weyl序列失去紧性。最后，在强场区域，重标度极小元趋近于极限泛函中最大化狄利克雷能量的极小元，同时拓扑荷和归一化能量测度集中在$H^{-1/4}$尺度上。
英文摘要
We study a fourth-order variational model for two-dimensional frustrated ferromagnets with competing exchange interactions and an applied magnetic field of strength $H>0$. For every $H>1/4$, we prove that the energy admits minimizers in the topological classes $Q=\pm1$ and that every minimizing sequence is precompact in $H^2$ modulo translations. The main difficulty is that spectral coercivity degenerates as $H\downarrow1/4$. Using a Helmholtz circle-mean identity, we prove that the residual energy of every nonzero-degree configuration has a uniform positive lower bound, even at the degenerate endpoint. Together with a sphere-valued $H^2$-splitting construction, this provides a threshold-stable binding inequality and yields compactness throughout the coercive regime. We also identify $H=1/4$ as the sharp spectral threshold. Below it the energy is unbounded from below, whereas at the threshold nonzero-degree configurations retain a positive energy barrier and degree-zero Weyl sequences lose compactness. Finally, in the strong-field regime, rescaled minimizers approach those minimizers of the limiting functional that maximize the Dirichlet energy, while topological-charge and normalized-energy measures concentrate on the $H^{-1/4}$-scale.
