On Capillary Free Surfaces without Gravity

Abstract

The non-parametric capillary problem in the absence of gravity can be replaced by a Dirichlet problem for a quasilmnear equation of mixed type by introducing a stream function. We prove a maximum principle for an expression depending on the gradient of the stream function and, furthermore, regularity properties of $C^1(\bar{\Omega }$-solutions in domains $\Omega$ with corners.