Abstract
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 noninterpenetration 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.
Highlights
Liquid films spanning rigid frames have been of longstanding interest to physicists and mathematicians, thanks to the sheer beauty of the countless observable shapes
The typical thickness of a soap film is on the submicron scale, while a bounding loop made of a strand of human hair would have a cross section with characteristic thickness of some tens of microns: this indicates that in many practical examples of liquid films bounded by flexible loops the characteristic thickness of the loop is at least two orders of magnitude greater than the thickness of the film
We introduced a mathematical model for experiments in which a thin filament in the form of a closed loop is spanned by a liquid film
Summary
Liquid films spanning rigid frames have been of longstanding interest to physicists and mathematicians, thanks to the sheer beauty of the countless observable shapes. First and foremost, De Lellis et al (2014) treat the surface as the support of a Radon measure, adopting a spatial point of view, and thereby obtain the optimal soap film regularity defined by Almgren (1976) and Taylor (1976) Their definition of the spanning conditions, built on ideas of Harrison (2014) [further developed by Harrison and Pugh (2016a, b)], allows for an apt treatment of the free-boundary problem, whereas all prior approaches become rather difficult to use when the boundary of the spanning surface is not prescribed. This is accomplished in Lemmas 3.4 and 3.5, and based on these results, we establish the lower semicontinuity property that is needed to establish, in Theorem 3.6, the existence of a solution to the Kirchhoff– Plateau problem
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