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

Ginster, Janusz; Zwicknagl, Barbara

doi:10.1007/s00332-022-09847-0

Cited by 7 publications

(4 citation statements)

References 51 publications

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“…[10,14] that show that there are parameter regimes in which uniform structures are preferred. A more detailed analysis of the minimal energy depending on the parameters is currently in preparation (see Ginster et al [15]).…”

Section: Model and The Main Resultsmentioning

confidence: 99%

“…A more detailed analysis of the minimal energy depending on the parameters is currently in preparation (see Ginster et al. [15]). In this paper, we wish to focus on the case of a large surface tension or a small coupling constant.…”

Section: Model and The Main Resultsmentioning

confidence: 99%

“…This in particular means that the formation of patterns is not expected in the regime −∞ < 𝑞 < 𝑞 * . We currently build on the heuristics presented in Section 2 and precisely identify the regimes in which the raft formation is expected in order to analyze the properties of minimizers in those cases (see Ginster et al [15]).…”

Section: Conclusion and Outline For Futurementioning

confidence: 99%

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Variational models for pattern formation in biomembranes—A Γ‐convergence result

Pešić

2023

Proc Appl Math and Mech

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Biological membranes are thin structures that are composed of various components. The different components often form microdomains, called lipid rafts, that are arranged in complex patterns. To explain this pattern formation, variational models based on Cahn–Hilliard‐type energies have been introduced that couple the local composition of the membrane to its local curvature, which renders the resulting functionals nonlocal. In this note, we review and extend recent qualitative results on related variational models. This includes a technique to deal with Neumann‐boundary conditions in the construction of a recovery sequence.

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Section: Model and The Main Resultsmentioning

confidence: 99%

Section: Model and The Main Resultsmentioning

confidence: 99%

Section: Conclusion and Outline For Futurementioning

confidence: 99%

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Variational models for pattern formation in biomembranes—A Γ‐convergence result

Pešić

2023

Proc Appl Math and Mech

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“…A mathematical analysis of micromagnetics models incorporating DMI terms can be found in [37,39,41]. We also refer to the works [16,17,18] for the discrete-to-continuum setting, as well as to [32] for a study of associated energy scaling laws, and to [35,24] for related numerical and dimension-reduction results.…”

Section: Introductionmentioning

confidence: 99%

Stochastic Homogenization of Micromagnetic Energies and Emergence of Magnetic Skyrmions

Davoli,

D’Elia,

Ingmanns

2024

J Nonlinear Sci

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We perform a stochastic-homogenization analysis for composite materials exhibiting a random microstructure. Under the assumptions of stationarity and ergodicity, we characterize the Gammalimit of a micromagnetic energy functional defined on magnetizations taking value in the unit sphere, and including both symmetric and antisymmetric exchange contributions. This Gamma-limit corresponds to a micromagnetic energy functional with homogeneous coefficients. We provide explicit formulas for the effective magnetic properties of the composite material in terms of homogenization correctors. Additionally, the variational analysis of the two exchange energy terms is performed in the more general setting of functionals defined on manifold-valued maps with Sobolev regularity, in the case in which the target manifold is a bounded, orientable smooth surface with tubular neighborhood of uniform thickness. Eventually, we present an explicit characterization of minimizers of the effective exchange in the case of magnetic multilayers, providing quantitative evidence of Dzyaloshinskii's predictions on the emergence of helical structures in composite ferromagnetic materials with stochastic microstructure.

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On the Formation of Microstructure for Singularly Perturbed Problems with Two, Three, or Four Preferred Gradients

Ginster

2024

J Nonlinear Sci

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In this manuscript, singularly perturbed energies with 2, 3, or 4 preferred gradients subject to incompatible Dirichlet boundary conditions are studied. This extends results on models for martensitic microstructures in shape memory alloys ($$N=2$$ N = 2 ), a continuum approximation for the $$J_1-J_3$$ J 1 - J 3 -model for discrete spin systems ($$N=4$$ N = 4 ), and models for crystalline surfaces with N different facets (general N). On a unit square, scaling laws are proved with respect to two parameters, one measuring the transition cost between different preferred gradients and the other measuring the incompatibility of the set of preferred gradients and the boundary conditions. By a change of coordinates, the latter can also be understood as an incompatibility of a variable domain with a fixed set of preferred gradients. Moreover, it is shown how simple building blocks and covering arguments lead to upper bounds on the energy and solutions to the differential inclusion problem on general Lipschitz domains.

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

Abstract: We study pattern formation in magnetic compounds near the helimagnetic/ferromagnetic transition point in case of Dirichlet boundary conditions on the spin field. The energy functional is a continuum approximation of a $$J_1-J_3$$ J 1 - J 3 model and was recently deriv… Show more

Cited by 7 publications

References 51 publications

Variational models for pattern formation in biomembranes—A Γ‐convergence result

Variational models for pattern formation in biomembranes—A Γ‐convergence result

Stochastic Homogenization of Micromagnetic Energies and Emergence of Magnetic Skyrmions

On the Formation of Microstructure for Singularly Perturbed Problems with Two, Three, or Four Preferred Gradients

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