  We consider the sloshing problem for an incompressible, inviscid,
irrotational fluid in an open container, including effects due to surface
tension on the free surface. We restrict ourselves to a constant contact angle
and seek time-harmonic solutions of the linearized problem, which describes the
time-evolution of the fluid due to a small initial disturbance of the surface
at rest. As opposed to the zero surface tension case, where the problem reduces
to a partial differential equation for the velocity potential, we obtain a
coupled system for the velocity potential and the free surface displacement. We
derive a new variational formulation of the coupled problem and establish the
existence of solutions using the direct method from the calculus of variations.
We prove a domain monotonicity result for the fundamental sloshing eigenvalue.
In the limit of zero surface tension, we recover the variational formulation of
the mixed Steklov-Neumann eigenvalue problem and give the first-order
perturbation formula for a simple eigenvalue.
