  We establish the existence of saddle points for a free boundary problem
describing the two-dimensional free surface of a ferrofluid which undergoes
normal field instability (also known as Rosensweig instability). The starting
point consists in the ferro-hydrostatic equations for the magnetic potentials
in the ferrofluid and air, and the function describing their interface. The
former constitute the strong form for the Euler-Lagrange equations of a
convex-concave functional. We extend this functional in order to include
interfaces that are not necessarily graphs of functions. Saddle points are then
found by iterating the direct method of the calculus of variations and by
applying classical results of convex analysis. For the existence part we assume
a general (arbitrary) non linear magnetization law. We also treat the case of a
linear law: we show, via convex duality arguments, that the saddle point is a
constrained minimizer of the relevant energy functional of the physical
problem.
