Kirchhoff-Plateau问题的解
Solution of the Kirchhoff-Plateau problem
摘要：
Kirchhoff-Plateau问题涉及一个系统的平衡形状，在该系统中，一个闭环形式的柔性细丝被一个液膜跨越，细丝被建模为Kirchhoff棒，跨越表面的作用仅仅是由于表面张力。我们建立了一个平衡形状的存在，最大限度地减少了系统的总能量下的物理约束的非相互渗透的物质，但允许点的表面上的边界循环接触。在我们的处理中，边界环保留了有限的横截面厚度和非零体积，而液体膜由有限的二维Hausdorff测度表示。此外，液体膜接触边界环的表面的区域不是先验规定的。我们的数学结果证实了所选模型的物理相关性。事实上，无论长丝的表面张力和弹性响应之间的竞争有多强，系统总是能够进行调整以实现符合实验中遇到的物理约束的配置。
英文摘要：
The Kirchhoff-Plateau problem concerns the equilibrium shapes of a system in which a flexible filament in the form of a closed loop is spanned by a liquid film, with the filament being modeled as a Kirchhoff rod and the action of the spanning surface being solely due to surface tension. We establish the existence of an equilibrium shape that minimizes the total energy of the system under the physical constraint of non-interpenetration of matter, but allowing for points on the surface of the bounding loop to come into contact. In our treatment, the bounding loop retains a finite cross-sectional thickness and a nonvanishing volume, while the liquid film is represented by a set with finite two-dimensional Hausdorff measure. Moreover, the region where the liquid film touches the surface of the bounding loop is not prescribed a priori. Our mathematical results substantiate the physical relevance of the chosen model. Indeed, no matter how strong is the competition between surface tension and the elastic response of the filament, the system is always able to adjust to achieve a configuration that complies with the physical constraints encountered in experiments.
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