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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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

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

First available in Project Euclid: 14 September 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]

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


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.

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