Consider a container separated into two subdomains: one is occupied by
an incompressible ferrofluid and the other is occupied by air (or
another fluid satisfying a linear magnetization law). Both regions are
subjected to a parallel, upwards pointing magnetic field of constant
strength H > 0. Ignore any transient phenomena that occur
when the field is turned on and assume a steady state, so that Maxwell’s
equations of magnetostatics are satisfied. Varying H, we obtain the so-called normal field (or Rosensweig) instability. It is a surface instability in the sense that, for small H the surface of a ferrofluid remains undisturbed and that for H > H_c
(a specific constant) static wave-like patterns appear on it. We study
this instability as a free interface problem, by considering an
appropriate two-phase action functional with unknown variables the
magnetic potential and the characteristic function of the set occupied
by the ferrofluid.