We establish the existence of spatially localised one-dimensional free surfaces of a ferrofluid near onset of the Rosensweig instability, assuming a general (nonlinear) magnetisation law. It is shown that the ferrohydrostatic equations can be derived from a variational principle that allows one to formulate them as an (infinite-dimensional) spatial Hamiltonian system in which the unbounded free-surface direction plays the role of time. A centremanifold reduction technique converts the problem for small solutions near onset to an equivalent Hamiltonian system with finitely many degrees of freedom. Normal-form theory yields the existence of homoclinic solutions to the reduced system, which correspond to spatially localised solutions of the ferrohydrostatic equations.

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In this work we show that the Kirchhoff-Love model for a hinged plate Δ 2 v = f in E, v = Δv − (1 − σ)κvn = 0 on ∂E admits, for f ∈ L 2 (E) and −1 < σ < 1, a unique weak solution in W 2,2 (E) ∩ W 1,2 0 (E) which satisfies the positivity preserving property when E ⊆ R 2 is a bounded, convex domain with C 2,1 boundary.