## Journal of the Mathematical Society of Japan

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

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

Miyuki KOISO and Bennett PALMER

#### 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