Rigorous existence of localized axisymmetric ferrofluid patterns

Prove the existence of localized axisymmetric steady‑state solutions (spikes/rings) to the ferrohydrostatic free‑surface problem with vertical magnetic field, modeled by the coupled Laplace equations and interface conditions in the ferrofluid system (equations \eqref{eqn:ferrofluidic}).

Background

Experiments in the Rosensweig instability show stable localized spikes and patches on ferrofluid surfaces under vertical magnetic fields, and numerical energy‑minimization captures axisymmetric and dihedral‑symmetric patches.

A rigorous existence proof for localized axisymmetric solutions of the full ferrohydrostatic interface problem has not yet been established.