Stability analysis for exotic containers

We consider the rotationally symmetric exotic containers constructed by R. Finn. These are containers which for a certain volume of fluid support a one-parameter family of equilibrium configurations all of which are rotationally symmetric and include in the family one whose free surface is a horizontal disk meeting the wall of the container at a prescribed contact; angle. The construction works for any gravitational field. The stability question was addressed by P. Concus and R. Finn who proved instability for extreme values of the Bond number with numerical calculations indicating instability in general. Here we prove analytically the instability result in all cases. AMS subject classification: 35J25, 49N10, 53A10.