On global minimizers for a variational problem with long-range interactions

Choksi, Rustum

doi:10.1090/s0033-569x-2012-01316-9

Cited by 14 publications

(10 citation statements)

References 66 publications

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“…Importantly, the model in (1.1) is a paradigm for the energy-driven pattern forming systems in which spatial patterns (global or local energy minimizers) form as a result of the competition of short-range attractive and long-range repulsive forces. This is why this model and its generalizations attracted considerable attention of mathematicians in recent years (see, e.g., [31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49], this list is certainly not exhaustive). In particular, the volume-constrained global minimization problem for (1.1) in the whole space with no neutralizing background, which we will also refer to as the "self-energy problem", has been investigated in [34,37,45,49].…”

Section: -3mentioning

confidence: 99%

“…From (5.42), (5.44) and (5.45) we then get 46) for ε small enough. Letting nowû ε (x) :=ũ ε (ε 1/3 x) be the rescaled function which satisfies 47) for every fixed ρ 0 > 0 and ε sufficiently small, we get 48) where Γ ρ 0 is defined via (5.21). Recalling Corollary 4.6 and choosing ρ 0 ≥ R 1 , we obtain 49) which gives (5.40) by first letting ρ → 0 and then δ → δ.…”

Section: Equidistribution Of Energymentioning

confidence: 99%

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Low Density Phases in a Uniformly Charged Liquid

Knüpfer

Muratov

Novaga

2016

Commun. Math. Phys.

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This paper is concerned with the macroscopic behavior of global energy minimizers in the three-dimensional sharp interface unscreened Ohta-Kawasaki model of diblock copolymer melts. This model is also referred to as the nuclear liquid drop model in the studies of the structure of highly compressed nuclear matter found in the crust of neutron stars, and, more broadly, is a paradigm for energy-driven pattern forming systems in which spatial order arises as a result of the competition of short-range attractive and long-range repulsive forces. Here we investigate the large volume behavior of minimizers in the low volume fraction regime, in which one expects the formation of a periodic lattice of small droplets of the minority phase in a sea of the majority phase.Under periodic boundary conditions, we prove that the considered energy Γ-converges to an energy functional of the limit "homogenized" measure associated with the minority phase consisting of a local linear term and a non-local quadratic term mediated by the Coulomb kernel. As a consequence, asymptotically the mass of the minority phase in a minimizer spreads uniformly across the domain. Similarly, the energy spreads uniformly across the domain as well, with the limit energy density minimizing the energy of a single droplets per unit volume. Finally, we prove that in the macroscopic limit the connected components of the minimizers have volumes and diameters that are bounded above and below by universal constants, and that most of them converge to the minimizers of the energy divided by volume for the whole space problem.

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Section: -3mentioning

confidence: 99%

Section: Equidistribution Of Energymentioning

confidence: 99%

Low Density Phases in a Uniformly Charged Liquid

Knüpfer

Muratov

Novaga

2016

Commun. Math. Phys.

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“…The sharp interface functional (1.3) has recently received attention from many authors; see, for example, [1,4,5,7,8,12,23,30]. It is a natural extension of the isoperimetric problem.…”

Section: Introductionmentioning

confidence: 99%

Nonexistence of a Minimizer for Thomas–Fermi–Dirac–von Weizsäcker Model

Otto

2013

Comm Pure Appl Math

118

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“…It to be pointed out is that the two fields in this description do not stand for cell types themselves, but any cell type can be described as a suitable linear combination of the two fields. The phase field literature is replete with formulations that model more complex structures (Choksi, 2012), including lipid bilayers (Dai and Promislow, 2013) and tubular structures (Kraitzman and Promislow, 2015). Rather than review those works this perspective has chosen to dwell on a few broader observations.…”

Section: Discussionmentioning

confidence: 99%

Perspectives on the mathematics of biological patterning and morphogenesis

Garikipati

2017

Journal of the Mechanics and Physics of Solids

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A central question in developmental biology is how size and position are determined. The genetic code carries instructions on how to control these properties in order to regulate the pattern and morphology of structures in the developing organism. Transcription and protein translation mechanisms implement these instructions. However, this cannot happen without some manner of sampling of epigenetic information on the current patterns and morphological forms of structures in the organism. Any rigorous description of space-and time-varying patterns and morphological forms reduces to one among various classes of spatio-temporal partial differential equations. Reaction-transport equations represent one such class. Starting from simple Fickian diffusion, the incorporation of reaction, phase segregation and advection terms can represent many of the patterns seen in the animal and plant kingdoms. Morphological form, requiring the development of three-dimensional structure, also can be represented by these equations of mass transport, albeit to a limited degree. The recognition that physical forces play controlling roles in shaping tissues leads to the conclusion that (nonlinear) elasticity governs the development of morphological form. In this setting, inhomogeneous growth drives the elasticity problem. The combination of reaction-transport equations with those of elasto-growth makes accessible a potentially unlimited spectrum of patterning and morphogenetic phenomena in developmental biology. This perspective communication is a survey of the partial differential equations of mathematical physics that have been proposed to govern patterning and morphogenesis in developmental biology. Several numerical examples are included to illustrate these equations and the corresponding physics, with the intention of providing physical insight wherever possible.

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On global minimizers for a variational problem with long-range interactions

Cited by 14 publications

References 66 publications

Low Density Phases in a Uniformly Charged Liquid

Low Density Phases in a Uniformly Charged Liquid

Nonexistence of a Minimizer for Thomas–Fermi–Dirac–von Weizsäcker Model

Perspectives on the mathematics of biological patterning and morphogenesis

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