具有拓扑奇点的二维离散能量极小化子的偶极子清除
Clearing-out of dipoles for minimisers of 2-dimensional discrete energies with topological singularities
AI总结：
针对含拓扑奇点的二维离散能量，提出组合式偶极子移除论证，利用最大流最小割与平面对偶刻画边界能量及通量对内部缺陷数的尖锐控制条件。
摘要：
在分析材料缺陷（例如自旋系统和超导体中的涡旋，或金属中的孤立位错）的离散模型时，一个关键问题是：关于一个区域边界能量的信息，是否足以控制内部缺陷的数量。我们针对一大类离散模型（包括 XY 系统和螺旋位错模型）提出了一种通用的组合式偶极子移除论证，使得能够证明尖锐条件：在这些条件下，受控通量和边界能量保证极小化子在内部具有零个或一个电荷。该论证结合使用最大流最小割定理与针对平面图的特设对偶，并且对于定义相互作用能量的函数的变化具有稳健性。
英文摘要：
A key question in the analysis of discrete models for material defects, such as vortices in spin systems and superconductors or isolated dislocations in metals, is whether information on boundary energy for a domain can be sufficient for controlling the number of defects in the interior. We present a general combinatorial dipole-removal argument for a large class of discrete models including XY systems and screw dislocation models, allowing to prove sharp conditions under which controlled flux and boundary energy guarantee to have minimizers with zero or one charges in the interior. The argument uses the max-flow min-cut theorem in combination with an ad-hoc duality for planar graphs, and is robust with respect to changes of the function defining the interaction energies.
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