Journal of the Mathematical Society of Japan

On a variational problem for soap films with gravity and partially free boundary

Miyuki KOISO and Bennett PALMER

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Abstract

We pose a variational problem for surfaces whose solutions are a geometric model for thin films with gravity which is partially supported by a given contour. The energy functional contains surface tension, a gravitational energy and a wetting energy, and the Euler-Lagrange equation can be expressed in terms of the mean curvature of the surface, the curvatures of the free boundary and a few other geometric quantities. Especially, we study in detail a simple case where the solutions are vertical planar surfaces bounded by two vertical lines. We determine the stability or instability of each solution.

Article information

Source
J. Math. Soc. Japan, Volume 57, Number 2 (2005), 333-355.

Dates
First available in Project Euclid: 14 September 2006

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1158242062

Digital Object Identifier
doi:10.2969/jmsj/1158242062

Mathematical Reviews number (MathSciNet)
MR2123236

Zentralblatt MATH identifier
1072.49029

Subjects
Primary: 49Q10: Optimization of shapes other than minimal surfaces [See also 90C90]
Secondary: 58E10: Applications to the theory of geodesics (problems in one independent variable) 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]

Keywords
variational problem stability of critical point partially free boundary soap film gravitational energy

Citation

KOISO, Miyuki; PALMER, Bennett. On a variational problem for soap films with gravity and partially free boundary. J. Math. Soc. Japan 57 (2005), no. 2, 333--355. doi:10.2969/jmsj/1158242062. https://projecteuclid.org/euclid.jmsj/1158242062


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