  The phase behavior of a large but finite Ising ferromagnet in the presence of
competing surface magnetic fields +/- H_s is studied by Monte Carlo simulations
and by phenomenological theory. Specifically, the geometry of a double pyramid
of height 2L is considered, such that the surface field is positive on the four
upper triangular surfaces of the bi-pyramid and negative on the lower ones. It
is shown that the total spontaneous magnetization vanishes (for L -> infinity)
at the temperature T_f(H), related to the "filling transition" of a
semi-infinite pyramid, which can be well below the critical temperature of the
bulk. The discontinuous vanishing of the magnetization is accompanied by a
susceptibility that diverges with a Curie-Weiss power law, when the transition
is approached from either side. A Landau theory with size-dependent critical
amplitudes is proposed to explain these observations, and confirmed by finite
size scaling analysis of the simulation results. The extension of these results
to other nanosystems (gas-liquid systems, binary mixtures, etc.) is briefly
discussed.
