二维向量值Allen-Cahn系统的边界渐近行为
Boundary asymptotics for two-dimensional vectorial Allen-Cahn systems
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中文总结 AI 辅助
本文研究二维向量值Allen-Cahn系统临界点的渐近性质，扩展Bethuel的平面内部理论至Neumann边界情形，通过极限应力-能量张量分析边界几何，明确了边界质量的相关规律。
中文摘要 AI 辅助
我们研究具有有限多个非退化势阱、满足齐次Neumann边界条件的二维向量值Allen-Cahn能量临界点的渐近性质。对于任意能量一致有界的序列，我们证明其极限全测度与势测度在边界附近支撑于闭的可数1-可求长集上，且满足偏差关系；势测度定义了一个自由边界稳定可求长变流形。边界质量可能出现：权重在正则边界弧上为常数，有限型接合点满足投影平衡，唯一的非边界分支正交交于边界。这为Bethuel的平面内部理论提供了Neumann边界下的扩展。核心思路是从极限应力-能量张量中读取边界几何，该张量即使在存在边界集中的情况下，也能将势测度识别为稳定界面测度。
英文摘要
We study asymptotic properties of critical points of the two-dimensional vectorial Allen-Cahn energy with finitely many non-degenerate wells, subject to a homogeneous Neumann boundary condition. For any sequence with uniformly bounded energy, we prove that the limiting full and potential measures are supported on a closed countably 1-rectifiable set up to the boundary and satisfy the discrepancy relations. The potential measure defines a free-boundary stationary rectifiable varifold. Boundary mass may occur: weights are constant on regular boundary arcs, finite-type junctions obey projected balance, and the sole non-boundary branch meets the boundary orthogonally. This provides a Neumann-boundary extension of Bethuel's planar interior theory. The key idea is to read the boundary geometry from the limiting stress-energy tensor, which identifies the potential-energy measure as the stationary interfacial measure even in the presence of boundary concentration.
发表机构
- School of Mathematical Sciences, Zhejiang University（浙江大学数学科学学院）
- School of Mathematical Sciences, Peking University（北京大学数学科学学院）
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