On some estimates for a fluid surface in a short capillary tube

In this note we investigate the problem of a fluid surface in a capillary tube of cross section Ω, which is assumed to be so short that the fluid rises to the top along the rim of the tube. Our main result is a minimum principle for an appropriate functional combination of the solution and its gradient. As an application of this minimum principle, we obtain some a priori estimates in terms of the curvature of ∂Ω. 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 ∂Ω.