The concertina pattern

Abstract

This is a continuation of a series of papers on the concertina pattern. The concertina pattern is a ubiquitous metastable, nearly periodic magnetization pattern in elongated thin film elements. In previous papers, a reduced variational model for this pattern was rigorously derived from 3-d micromagnetics. Numerical simulations of the reduced model reproduce the concertina pattern and show that its optimal period w opt is an increasing function of the applied external field h ext . The latter is an explanation of the experimentally observed coarsening. Domain theory, which can be heuristically derived from the reduced model, predicts and quantifies this dependence of w opt on h ext . In this paper, we rigorously extract these heuristic observations of domain theory directly from the reduced model. The main ingredient of the analysis is a new type of estimate on solutions of a perturbed Burgers equation.

Energy Scaling Law for a Singularly Perturbed Four-Gradient Problem in Helimagnetism

“…We point out that in particular a variety of magnetization patterns (including branching structures) have been successfully explained via scaling laws of continuum micromagnetic energies, see, e.g., , , Dabade et al (2019), DeSimone et al (2006a), DeSimone et al (2006b), Knüpfer and Muratov (2011), Otto and Steiner (2010), Otto and Viehmann (2010), Venkatraman et al (2020)). While these models typically contain local and non-local terms, we will focus on a purely local model that arises -at least heuristically -from a frustrated spin system, see Diep (2013), Diep (2015) for the general context and Cicalese and Solombrino (2015), Cicalese et al (2019) for the specific setting considered here.…”

Variational methods for a singular SPDE yielding the universality of the magnetization ripple

“…+∞, otherwise. 25 Here we understand 𝐸 𝑟𝑒𝑛 (𝑣, 𝐹; ⋅) as a measurable function of 𝜉, which is a composition of the measurable function 𝜉 ↦ (𝜉, 𝑣, 𝐹) and the continuous function (𝜉, 𝑣, 𝐹) ↦ 𝐸 𝑟𝑒𝑛 (𝑣, 𝐹; ⋅).…”

“…Hence, the push-forward (𝐸 𝑟𝑒𝑛 ) # ⟨⋅⟩ lif t is well-defined as a probability measure on the space of lower semicontinuous functionals equipped with the Borel 𝜎-algebra corresponding to the topology of Γ-convergence (based on the strong 𝐿 2 -topology). We now define ⟨⋅⟩ ext as the joint law of 𝜉 and 𝐸 𝑟𝑒𝑛 (𝑣, 𝐹; ⋅) 25. (i) This is immediate by the definition of ⟨⋅⟩ ext and the fact that by Proposition 1.9, ⟨⋅⟩ is supported on the Hölder space …”

A smectic liquid crystal model in the periodic setting

“…In this paper, we approach the compactness via a different argument in the periodic setting. Our proof is motivated by recent work on related variational models in the periodic setting [5,10,12,20,21] where strong convergence of a weakly convergent L 2 sequence is proved via estimates on Fourier series. Given a sequence u ε weakly converging in L 2 (T 2 ), to prove strong convergence of u ε in L 2 , it is sufficient to show that there is no concentration in the high frequencies.…”