# A free boundary approach to the Rosensweig instability of ferrofluids

## TL;DR

This paper proves the existence of saddle points for a free boundary problem modeling the Rosensweig instability in ferrofluids, using variational methods and convex analysis, accommodating general nonlinear magnetization laws.

## Contribution

It extends the variational framework to include non-graph interfaces and establishes existence results for saddle points under general magnetization laws.

## Key findings

- Existence of saddle points for the free boundary problem.
- Extension of the functional to non-graph interfaces.
- Convex duality used for linear magnetization law case.

## Abstract

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.