A minimum principle for a soap film problem in $${\mathbb{R}^{2}}$$

Abstract

In this note, we investigate the problem of a thin extensible film (a soap film), under the influence of gravity and surface tension, supported by the contour of a given strictly convex smooth domain Ω. Our main result is a minimum principle for an appropriate combination of u(x) and \({\left\vert \nabla u\left( \mathbf{x}\right) \right\vert }\) , that is, a kind of P-function in the sense of Payne (see the book of Sperb in Maximum Principles and Their Applications. Academic Press, New York, 1981), where u(x) is the solution of our problem. As an application of this minimum principle, we obtain some a priori estimates for the surface represented by the thin extensible film, in terms of the curvature of \({\partial \Omega}\) . The proofs make use of Hopf’s maximum principles, some topological arguments regarding the local behavior of analytic functions and some computations in normal coordinates with respect to the boundary \({\partial \Omega }\).