Striped patterns for generalized antiferromagnetic functionals with power law kernels of exponent smaller than $d+2$

Kerschbaum, Alicja

doi:10.48550/arxiv.2101.02992

Cited by 3 publications

(7 citation statements)

References 35 publications

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“…As shown in [Ker21], the above stability argument can be extended to all p > d + 1, provided τ is sufficiently small depending on p.…”

Section: Preliminary Lemmasmentioning

confidence: 87%

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Periodic striped configurations in the large volume limit

Daneri¹,

Runa²

2021

Preprint

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We show striped pattern formation in the large volume limit for a class of generalized antiferromagnetic local/nonlocal interaction functionals in general dimension previously considered in [GR19; DR19; DR21a] and in [GLL06; GS16] in the discrete setting. In such a model the relative strength between the short range attractive term favouring pure phases and the long range repulsive term favouring oscillations is modulated by a parameter τ . For τ < 0 minimizers are trivial uniform states. It is conjectured that ∀ d ≥ 2 there exists 0 < τ ≪ 1 such that for all 0 < τ ≤ τ and for all L > 0 minimizers are striped/lamellar patterns. In [DR19] the authors prove the above for L = 2kh * τ , where k ∈ N and h * τ is the optimal period of stripes for a given 0 < τ ≤ τ . The purpose of this paper is to show the validity of the conjecture for generic L.

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“…As shown in [Ker21], the above stability argument can be extended to all p > d + 1, provided τ is sufficiently small depending on p.…”

Section: Preliminary Lemmasmentioning

confidence: 87%

“…In [Ker21] such a characterization of minimizers was proved to hold also in a small open range of exponents below d + 2. We mention that in physical applications the power law interactions have exponents smaller than or equal to d + 1.…”

Section: Scientific Contextmentioning

confidence: 92%

Periodic striped configurations in the large volume limit

Daneri¹,

Runa²

2021

Preprint

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“…Regarding the limit functional (2.1), we recall the following result, obtained in [11] for p ≥ d + 2 and extended to a range of exponent below d + 2 in [19].…”

Section: Notation and Preliminariesmentioning

confidence: 99%

“…where E ⊂ R d , d ≥ 2, one-dimensionality and periodicity of minimizers in the thermodynamic limit has been proved in [15] in the discrete setting (for exponents p > 2d) and in [11] in the continuous one (for exponents p ≥ d + 2). In [19] the results of [11] have been recently extended to a small range of exponents below p = d + 2.…”

Section: Scientific Contextmentioning

confidence: 99%

“…For the most physically relevant exponents such as p = d + 1 (thin magnetic films), p = d (3D micromagnetics) and p = d − 2 (diblock copolymers), a rigorous proof of pattern formation in dimension d ≥ 2 is still a challenging open problem. The main difficulty, which in [15], [17], [11] and [19] is resolved for higher exponents p, is to prove that symmetry breaking occurs, namely that minimizers of (1.7) have less symmetries than the functional itself. In this case symmetry under coordinate permutations is lost, but if one considers the functional analogous to (1.7) where the 1-norm is substituted by the Euclidean norm, full rotational symmetry loss is expected to occur as well.…”

Section: Scientific Contextmentioning

confidence: 99%

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One-dimensionality of the minimizers in the large volume limit for a diffuse interface attractive/repulsive model in general dimension

Daneri¹,

Runa²

2021

Preprint

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In this paper we consider the diffuse interface generalized antiferromagnetic model with local/nonlocal attractive/repulsive terms in competition studied in [9]. The parameters of the model are denoted by τ and ε: the parameter τ represents the relative strength of the local term with respect to the nonlocal one, while the parameter ε describes the transition scale in the Modica-Mortola type term. Restricting to a periodic box of size L, with L multiple of the period of the minimal one-dimensional minimizers, in [9] the authors prove that in any dimension d ≥ 1 and for small but positive τ and ε (eventually depending on L), the minimizers are nonconstant one-dimensional periodic functions. In this paper we prove that periodicity and onedimensionality of minimizers occurs also in the zero temperature analogue of the thermodynamic limit, namely as L → +∞.

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Exact periodic stripes for a local/nonlocal minimization problem with volume constraint

Daneri¹,

Runa²

2021

Preprint

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We consider a class of generalized antiferromagnetic local/nonlocal interaction functionals in general dimension, where a short range attractive term of perimeter type competes with a long range repulsive term characterized by a reflection positive power law kernel. Breaking of symmetry with respect to coordinate permutations and pattern formation for functionals in this class have been shown in [GR19; DR19b] and previously by [GS16] in the discrete setting, for a smaller range of exponents. Global minimizers of such functionals have been proved in [DR19b] to be given by periodic stripes of volume density 1/2 in any cube having optimal period size, also in the large volume limit. In this paper we study the minimization problem with arbitrarily prescribed volume constraint α ∈ (0, 1). We show that, in the large volume limit, minimizers are periodic stripes of volume density α, namely stripes whose one-dimensional slices in the direction orthogonal to their boundary are simple periodic with volume density α in each period. Results of this type in the one-dimensional setting, where no symmetry breaking occurs, have been previously obtained in [Mül93; AM01; RW03; CO05; GLL09].

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Striped patterns for generalized antiferromagnetic functionals with power law kernels of exponent smaller than $d+2$

Cited by 3 publications

References 35 publications

Periodic striped configurations in the large volume limit

Periodic striped configurations in the large volume limit

One-dimensionality of the minimizers in the large volume limit for a diffuse interface attractive/repulsive model in general dimension

Exact periodic stripes for a local/nonlocal minimization problem with volume constraint

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