Local energy minimality of capillary surfaces in the presence of symmetry

Abstract

Consider the stationary capillary problem of a drop of liquid attached to a fixed surface, so that the drop minimizes an energy functional subject to a volume constraint. There are many such capillary problems in which, due to the symmetry of the fixed surface, one cannot hope for a capillary surface which is a strict local minimum for energy. A weaker concept which is sensible to consider is that of local minimality modulo the isometries of space which map the fixed surface into itself. In other words, it is reasonable to attempt to show that, given a capillary surface, any nearby comparison surface will have energy greater than or equal to the given surface, and if the energy is equal, the comparison surface is simply a translation or rotation of the given surface. Eigenvalue conditions are derived which imply that a capillary surface is a strict local minimum modulo isometries, and are applied to the specific example of a liquid bridge between two parallel planes.

Liquid Bridges Between Balls: The Small Volume Instability

“…For fixed m, (4.6), (4.7) form a standard Sturm-Liouville problem, so it is natural to label the eigenvalues as λ jm . As in [12], we have λ 0m < λ 1m < λ 2m < · · · and λ 00 < λ 01 < λ 02 < · · · , and eigenfunctions corresponding to m = 0 are radially symmetric. A crucial part of the argument in [14] depended on the sign of λ 01 in different circumstances.…”