AI 中文总结
研究具有球值约束的 N 场系统变分模型，分析惩罚参数趋于无穷时极小值的渐近行为与正则性。通过建立二分法，对 N = 2 和 N≥3 情况分别研究，还推导极限能量表征等，扩展单场模型结果到多场设置。
AI 中文摘要
本文研究了一个变分模型，该模型通过惩罚项描述了 N 个耦合球值场的系统，该惩罚项施加了它们的和与规定球值映射一致的约束。我们分析了惩罚参数趋于无穷时极小值的渐近行为和正则性。根据相互作用场的数量建立了一个基本二分法。对于 N = 2，拓扑障碍可能会阻止精确球值分解的存在，导致大惩罚 regime 中最小能量的爆炸。相比之下，对于 N≥3，精确分解总是存在，产生均匀能量界并允许通过 Γ-收敛识别极限约束问题。我们通过将可允许配置分解为平均场和波动分量进一步推导出极限能量的显式表征。最后，我们建立了一个间隙现象，表明在适当的拓扑假设下，对于足够大的惩罚参数值，每个极小值必然是奇异的。这些结果突出了拓扑在耦合球值变分系统的渐近行为和正则性中的基本作用，并将单场模型的先前结果扩展到多场设置。
英文摘要
This paper investigates a variational model that describes a system of N coupled sphere-valued fields through a penalization term imposing the constraint that their sum coincides with a prescribed sphere value map. We analyze the asymptotic behavior and regularity of minimizers as the penalization parameter tends to infinity. A fundamental dichotomy is established according to the number of interacting fields. For N = 2, topological obstructions may prevent the existence of exact sphere-valued decompositions, leading to the blow-up of the minimal energy in the large-penalization regime. In contrast, for N $\ge$ 3, exact decompositions always exist, yielding uniform energy bounds and allowing the identification of the limiting constrained problem via $Γ$-convergence. We further derive an explicit characterization of the limiting energy by decomposing admissible configurations into an average field and fluctuation components. Finally, we establish a gap phenomenon showing that, under suitable topological assumptions, every minimizer is necessarily singular for sufficiently large values of the penalization parameter. These results highlight the fundamental role of topology in the asymptotic behavior and regularity of coupled sphere-valued variational systems and extend previous results on single-field models to the multi-field setting.
