Well known first order
necessary conditions for a liquid mass to be in equilibrium in contact with a
fixed solid surface declare that the free surface interface has mean curvature
prescribed in terms of the bulk accelerations acting on the liquid and meets the
solid surface in a materially dependent contact angle. We derive first order
necessary conditions for capillary surfaces in equilibrium in contact with solid
surfaces which may also be allowed to move. These conditions consist of
the same prescribed mean curvature equation for the interface, the same
prescribed contact angle condition on the boundary, and an additional integral
condition which may be said to involve, somewhat surprisingly, only the wetted
region.
An example of the kind of system under consideration is that of a floating ball in
a fixed container of liquid. We apply our first order conditions to this particular
problem.
Keywords
calculus of variations, capillarity, minimal surfaces,
constant mean curvature
