In this work we study a deformation of the minimal interface of two fluids in a vertical tube under the presence of gravitation. We show that a symmetry of the base of tube let us to apply a method developed earlier by the first author and based on the Crandall-Rabinowitz bifurcation theorem. Using the natural symmetry of the corresponding variational problem defined by a symmetry of region and restricting the functional to spaces of invariant functions we show the existence of bifurcation, and describe its local picture, for interfaces parametrized by the square and disc.
"Bifurcation of the equivariant minimal interfaces in a hydromechanics problem." Abstr. Appl. Anal. 1 (3) 291 - 304, 1996. https://doi.org/10.1155/S1085337596000152