Nonuniqueness and Uniqueness of Capillary Surfaces

It is shown that for any gravity field g and contact angle 7, an axially symmetric container can be found that differs arbitrarily little from a circular cylinder and can be half filled with liquid in a continuum of distinct ways, such that no two of the surface interfaces are mutually congruent and such that all of them are in equilibrium with the same mechanical energy. This answers affirmatively a question raised by Gulliver and Hildebrandt [1], who obtained such a container in the particular case g = 0, γ = π/2.For a particular surface in the continuum, it is shown that the second variation of energy can be made negative by a non-axisymmetric perturbation under the volume constraint, and thus that the surface can be embedded in a one-parameter family of nonsymmetric surfaces bounding constant volume, with decreasing energy. As a consequence, a rotationally symmetric container deviating arbitrarily little from a circular cylinder is characterized, so that an energy minimizing configuration filling half the container exists but will not be symmetric.Finally, conditions on the container curvature are given under which the symmetric configurations with prescribed volume are uniquely determined. In the special case for which the container is a sphere and g = 0, the symmetric configuration is unique and energy minimizing among all possible configurations.Details of this-work are given in [2] and in [3].