Symmetric Capillary Surfaces in a Cube Part 3. More Exotic Surfaces, Gravity

Previous numerical experiments are extended to more complex surfaces and in some cases to nonzero gravity. 1. The Mathematical Problem Let be the unit cube in R 3. We are considering subdomains having a piecewise smooth boundary @ = ? , where ? is a subset of the interior of and is a subset of the boundary @ of. We are looking for those subdomains which solve the following variational problem: The energy functional E = Z ? d? + b Z x 3 d ? cos # Z d is minimal under the restriction that the volume V = Z d attaines a prescribed value. It is a well known fact-going back to K. F. Gauu-that solutions of this variational problem must be such that the capillary surface ? has prescribed mean curvature 2H = bx 3 + p and the contact angle between ? and is equal to # (see, e.g., 5]). The mean curvature is thus constant if the Bond number b is zero. The number p is the Lagrange multiplier of the variational problem. It turns out that it is equal to the diierence of the pressures of two liquids ((uid or gas) occupying the domains and n , resp. In the following we assume that one part is lled with a uid and the other is void. This work is an extension of our earlier work 7] 8]. In 7] a long list of diierent conngurations was given. All surfaces shared several of the symmetries with the cube. In particular, graphs were presented in which energy E, total area A = R ? d?, and pressure p were plotted as functions of the volume for the entire