Second Order Expansion for the Nonlocal Perimeter Functional

We derive a macroscopic limit for a sharp interface version of a model proposed in Komura et al. (Langmuir 22:6771–6774, 2006) to investigate pattern formation due to competition of chemical and mechanical forces in biomembranes. We identify sub- and supercritical parameter regimes and show with the introduction of the autocorrelation function that the ground state energy leads to the isoperimetric problem in the subcritical regime, which is interpreted to not form fine scale patterns.

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$$\Gamma $$–limit for a sharp interface model related to pattern formation on biomembranes

Brazke

Knüpfer

Marciniak–Czochra

2023

Calc. Var.

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Onset of pattern formation in thin ferromagnetic films with perpendicular anisotropy

Brietzke

Knüpfer

2023

Calc. Var.

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We consider the onset of pattern formation in an ultrathin ferromagnetic film of the form $$\Omega _t:= \Omega \times [0,t]$$ Ω t : = Ω × [ 0 , t ] for $$\Omega \Subset \mathbb {R}^2$$ Ω ⋐ R 2 with preferred perpendicular magnetization direction. The relative micromagnetic energy is given by $$\begin{aligned} \mathcal {E}[M]= \int _{\Omega _t} d^2 |\nabla M|^2+ Q \int _{\Omega _t} (M_1^2+M_2^2) + \int _{\mathbb {R}^3} |\mathcal {H}(M)|^2 - \int _{\mathbb {R}^3} |\mathcal {H}(e_3 \chi _{\Omega _t})|^2, \end{aligned}$$ E [ M ] = ∫ Ω t d 2 | ∇ M | 2 + Q ∫ Ω t ( M 1 2 + M 2 2 ) + ∫ R 3 | H ( M ) | 2 - ∫ R 3 | H ( e 3 χ Ω t ) | 2 , describing the energy difference for a given magnetization $$M: \mathbb {R}^3 \rightarrow \mathbb {R}^3$$ M : R 3 → R 3 with $$|M| = \chi _{\Omega _t}$$ | M | = χ Ω t and the uniform magnetization $$e_3 \chi _{\Omega _t}$$ e 3 χ Ω t . For $$t \ll d$$ t ≪ d , we derive the scaling of the minimal energy and a BV-bound in the critical regime, where the base area of the film has size of order $$|\Omega |^{{\frac{1}{2}}} \sim (Q-1)^{{-\frac{1}{2}}} d e^{\frac{2\pi d}{t} \sqrt{Q-1}}$$ | Ω | 1 2 ∼ ( Q - 1 ) - 1 2 d e 2 π d t Q - 1 . We furthermore investigate the onset of non-trivial pattern formation in the critical regime depending on the size of the rescaled film.

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Second Order Expansion for the Nonlocal Perimeter Functional

Cited by 2 publications

References 37 publications

$$\Gamma $$–limit for a sharp interface model related to pattern formation on biomembranes

$$\Gamma $$–limit for a sharp interface model related to pattern formation on biomembranes

Onset of pattern formation in thin ferromagnetic films with perpendicular anisotropy

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