带边界界面的二维Allen-Cahn理论
A two-dimensional Allen-Cahn theory for interfaces with boundary
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中文总结 AI 辅助
本文在Fröhlich与Struwe的线丛框架下发展带边界界面的二维Allen-Cahn理论，构造边界模型解及Allen-Cahn截面，需用Lyapunov-Schmidt约化并克服边界带来的误差控制难题。
中文摘要 AI 辅助
我们在Fröhlich和Struwe引入的线丛框架下，发展了带边界界面的二维Allen-Cahn理论。核心新对象是穿孔平面上的模型解，其节点集为半直线。在界面边界附近，该解起到经典内部理论中异宿剖面的作用。但与内部情形不同，内部情形中垂直于界面方向的分析可简化为异宿满足的常微分方程，而边界模型具有非平坦水平集，必须作为真正的二维椭圆解来研究。本文第一部分，我们构造该模型解并建立其稳定性与可逆性理论。第二部分呈现主要应用：我们构造Allen-Cahn截面，其节点集集中于平面内任意给定的有限条不相交线段。该构造使用Lyapunov-Schmidt约化，但边界引入新困难：沿界面内部产生大误差项，控制这些项需要精细的假设和超出标准内部Allen-Cahn理论的新拼接论证。
英文摘要
We develop a two-dimensional Allen-Cahn theory for interfaces with boundary in the line-bundle framework introduced by Fröhlich and Struwe. The central new object is a model solution on the punctured plane whose nodal set is a half-line. Near the boundary of an interface, this solution plays the role of the heteroclinic profile in the classical interior theory. However, unlike the interior setting, where the analysis in directions normal to the interface reduces to the ODE satisfied by the heteroclinic, the boundary model has non-flat level sets and must be studied as a genuinely two-dimensional elliptic solution. In the first part of this paper, we construct this model solution and develop its stability and invertibility theory. In the second part of this paper, we present the main application: we construct Allen--Cahn sections whose nodal sets concentrate on any prescribed finite collection of disjoint line segments in the plane. The construction uses a Lyapunov-Schmidt reduction, but the boundary introduces a new difficulty: it creates large error terms along the interior of the interface. Controlling these terms requires a refined ansatz and new gluing arguments beyond the standard interior Allen-Cahn theory.
