Non uniqueness and uniqueness of capillary surfaces

Abstract

It is shown that for any gravity fieldg and contact angleγ, a containerC can be found which can be partially filled with liquid in a continuum of distinct ways, so as to obtain a one parameter familyF of capillary surface interfaces, no two of which are mutually congruent, and all of which bound the same liquid volume and yield the same mechanical energy. This answers affirmatively a question raised by Gulliver and Hildebrandt, who obtained such a container in the caseg=0,γ=π/2. For a particular configuration inF, the second variation of energy is calculated and it is shown that-at least for small g- it can be made negative. 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 cannot be symmetric. Finally, a condition on rotationally symmetric container shapes is given, for the existence of a unique disk-type symmetric stationary surface. In the particular case of a spherical container in zero gravity, this surface is unique and minimizing among all disk-type surfaces with the given contact angle and enclosed volume.