2022/05/20 by Brietzke, Birger, Knüpfer, Hans
#FOS: Physical sciences #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.2205.10061
We consider the onset of pattern formation in an ultrathin ferromagnetic film of the form Ωt := Ω × [0,t] for Ω \Subset ℝ2 with preferred perpendicular magnetization direction. The relative micromagnetic energy is given by E[M] amp;= ∫Ωt d2 |∇ M|2+ Q ∫Ωt (M12+M22) + ∫ℝ3 |H(M)|2 - ∫ℝ3 |H(e3 χ_ Ω)|2, describing the energy difference for a given magnetization M : ℝ3 → ℝ3 with |M| = χ_ Ωt and the uniform magnetization e3 χ_ Ωt. For t ≪ d, we establish the scaling of the energy and a BV-bound in the critical regime here the base area of the film is of order | Ω| ∼ (Q-1)1/2 d e^\frac2πdt √(Q-1). We furthermore investigate the onset of non-trivial pattern formation in the critical regime depending on the size of the rescaled film.