A variational approach to the study of capillary phenomena

The problem of determining the free surface of a liquid in a capillary tube, and of a liquid drop, sitting first on a horizontal plane and then on more general surfaces is considered. With some modifications, the method applies to the study of pendent drops and of rotating drops as well. The standard capillary problem, i.e. the determination of the free surface of a liquid in a thin tube of general cross section, which resuls from the simultaneous action of surface tension, boundary adhesion and gravity is discussed. It turns out that in this case the existence of the solution surface depends heavily on the validity of a simple geometric condition about the mean curvature of the boundary curve of the cross section of the capillary tube. Some particular examples of physical interest are also be discussed. Liquid drops sitting on or hanging from a fixed horizontal plane are discussed. The symmetry of the solutions (which can actually be proved, as consequence of a general symmetrization argument) now plays the chief role in deriving both the existence and the regularity of energy-minimizing configurations. When symmetry fails (this is the case, for example, when the contact angle between the drop and the plate is not constant, or when the supporting surface is not itself symmetric), then more sophisticated methods must be used. Extensions in this direction are outlined.