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.
"On a variational problem for soap films with gravity and partially free boundary." J. Math. Soc. Japan 57 (2) 333 - 355, April, 2005. https://doi.org/10.2969/jmsj/1158242062