当 $p$ 趋于 $2$ 时具有有限基本群的三维极小化 $p$-调和映射的极限行为
Limiting behavior of minimizing $p$-harmonic maps in 3d as $p$ goes to $2$ with finite fundamental group
AI总结：
本文研究三维空间中极小化 $p$-调和映射在 $p$ 趋于 2 时的极限行为，证明奇异集为有限长度直线段并集且极小化质量，并给出极限调和映射的局部与全局估计。
摘要：
我们研究从有界 Lipschitz 区域 $Ω\subset \mathbb{R}^{3}$ 到无边界的紧连通黎曼流形、且具有有限基本群的极小化 $p$-调和映射在 $p \nearrow 2$ 时的极限行为。我们证明存在一个长度有限的闭集 $S_{*}$，使得极小化 $p$-调和映射在 $Ω\setminus S_{*}$ 中收敛到局部极小化调和映射。我们证明在 $Ω$ 内部局部地看，奇异集 $S_{*}$ 是直线段的有限并集，并且它在适当的可容许链类中极小化质量。此外，我们对极限奇异调和映射建立了局部和全局估计。在额外假设下，我们证明在 $\overlineΩ$ 中全局地看，集合 $S_{*}$ 是直线段的有限并集，并且它在由给定边界数据和 $Ω$ 定义的适当可容许链类中极小化质量。
英文摘要：
We study the limiting behavior of minimizing $p$-harmonic maps from a bounded Lipschitz domain $Ω\subset \mathbb{R}^{3}$ to a compact connected Riemannian manifold without boundary and with finite fundamental group as $p \nearrow 2$. We prove that there exists a closed set $S_{*}$ of finite length such that minimizing $p$-harmonic maps converge to a locally minimizing harmonic map in $Ω\setminus S_{*}$. We prove that locally inside $Ω$ the singular set $S_{*}$ is a finite union of straight line segments, and it minimizes the mass in the appropriate class of admissible chains. Furthermore, we establish local and global estimates for the limiting singular harmonic map. Under additional assumptions, we prove that globally in $\overlineΩ$ the set $S_{*}$ is a finite union of straight line segments, and it minimizes the mass in the appropriate class of admissible chains, which is defined by a given boundary datum and $Ω$.
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