# Bounds on the energy of a soft cubic ferromagnet with large magnetostriction

## TL;DR

This paper establishes lower bounds on the micromagnetic energy of cubic ferromagnets with significant magnetostriction, explaining complex pattern formations in Galfenol disks through energetic analysis.

## Contribution

It provides ansatz-free lower bounds for the micromagnetic energy including magnetostriction in 2D samples, advancing understanding of pattern formation in ferromagnetic materials.

## Key findings

- Derived lower bounds for micromagnetic energy with magnetostriction.
- Explained zig-zag Landau patterns in Galfenol disks energetically.
- Connected pattern formation to energy minimization principles.

## Abstract

We complete the analysis initiated in [5] on the micromagnetics of cubic ferromagnets in which the role of magnetostriction is significant. We prove ansatz-free lower bounds for the scaling of the total micromagnetic energy including magnetostriction contribution, for a two-dimensional sample. This corresponds to the micromagnetic energy-per-unit-length of an infinitely thick sample. A consequence of our analysis is an explanation of the multi-scale zig-zag Landau state patterns recently reported in single crystal Galfenol disks from an energetic viewpoint. Our proofs combines a number of well-developed techniques in energy-driven pattern formation with a recent regularity result for the kinetic formulation of the Eikonal equation of Ghiraldin and Lamy.