  We provide, in the setting of Gauss' capillarity theory, a rigorous
derivation of the equilibrium law for the three dimensional structures known as
Plateau borders which arise in "wet" soap films and foams. A key step in our
analysis is a complete measure-theoretic overhaul of the homotopic spanning
condition introduced by Harrison and Pugh in the study of Plateau's laws for
two-dimensional area minimizing surfaces ("dry" soap films). This new point of
view allows us to obtain effective compactness theorems and energy
representation formulae for the homotopic spanning relaxation of Gauss'
capillarity theory which, in turn, lead to prove sharp regularity properties of
energy minimizers. The equilibrium law for Plateau borders in wet foams is also
addressed as a (simpler) variant of the theory for wet soap films.
