Estimates for the maxima of the moduli of the gradients for solutions of capillarity problems

One establishes a priori estimates for the maxima of the moduli of the gradients of the solutions for a class of nonlinear boundary-value problems, related to the description of capillarity phenomena. The variational formulation of such problems consists in the minimization of the energy functional of the fluid in the capillary tube, representing a cylinder in a curvilinear system of coordinates, in which the metric tensor has a special structure.