Asymmetric and Symmetric Double Bubbles in a Ternary Inhibitory System

Ren, Xiaofeng; Wei, Juncheng

doi:10.1137/140955720

Cited by 11 publications

(4 citation statements)

References 36 publications

(54 reference statements)

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“…Related to results in the present paper are the works, for κ = 0, of Ren and Wei in their series of papers [40][41][42][43][44][45][46][47][48][49][50]. Among many others, they build equilibrium patterns of type lamellar, ring/cylindrical, and spot/spherical solutions.…”

Section: Introduction and Main Resultssupporting

confidence: 54%

Periodic patterns for a model involving short-range and long-range interactions

Fall¹

2017

Preprint

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Section: Introduction and Main Resultssupporting

confidence: 54%

Periodic patterns for a model involving short-range and long-range interactions

Fall¹

2017

Preprint

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“…Double bubble like stationary sets and stationary assemblies are found in [23,24]. These sets do not meet the domain boundary.…”

Section: Discussionmentioning

confidence: 97%

The Impact of the Domain Boundary on an Inhibitory System: Existence and Location of a Stationary Half Disc

Ren

Shoup

2015

Commun. Math. Phys.

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The nonlocal geometric variational problem derived from the Ohta-Kawasaki diblock copolymer theory is an inhibitory system with self-organizing properties. The free energy, defined on subsets of a prescribed measure in a domain, is a sum of a local perimeter functional and a nonlocal energy given by the Green's function of Poisson's equation on the domain with the Neumann boundary condition. The system has the property of preventing a disc from drifting towards the domain boundary. This raises the question of whether a stationary set may have its interface touch the domain boundary. It is proved that a small, perturbed half disc exists as a stable stationary set, where the circular part of its boundary is inside the domain, as the interface, and the almost flat part of its boundary coincides with part of the domain boundary. The location of the half disc depends on two quantities: the curvature of the domain boundary, and a remnant of the Green's function after one removes the fundamental solution and a reflection of the fundamental solution. This reflection is defined with respect to any sufficiently smooth domain boundary. It is an interesting new concept that generalizes the familiar notions of mirror image and circle inversion. When the nonlocal energy is weighted less against the local energy, the stationary half disc sits near a maximum of the curvature; when the nonlocal energy is weighted more, the half disc appears near a minimum of the remnant function. There is also an intermediate case where the half disc is near a minimum of a combination of the curvature and the remnant function.

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“…The first is an assembly of perturbed double bubbles (the first plot in Figure 1) found by Ren and Wei in [29]. The special case of a single, perturbed double bubble stationary point was found earlier in [26,27]. One dimensional stationary points were found in [20,5].…”

mentioning

confidence: 81%

A stationary core-shell assembly in a ternary inhibitory system

Ren¹,

Wu²

2017

Discrete &Amp; Continuous Dynamical Systems - A

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A ternary inhibitory system motivated by the triblock copolymer theory is studied as a nonlocal geometric variational problem. The free energy of the system is the sum of two terms: the total size of the interfaces separating the three constituents, and a longer ranging interaction energy that inhibits micro-domains from unlimited growth. In a particular parameter range there is an assembly of many core-shells that exists as a stationary set of the free energy functional. The cores form regions occupied by the first constituent of the ternary system, the shells form regions occupied by the second constituent, and the background is taken by the third constituent. The constructive proof of the existence theorem reveals much information about the core-shell stationary assembly: asymptotically one can determine the sizes and locations of all the core-shells in the assembly. The proof also implies a kind of stability for the stationary assembly.

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Asymmetric and Symmetric Double Bubbles in a Ternary Inhibitory System

Cited by 11 publications

References 36 publications

Periodic patterns for a model involving short-range and long-range interactions

Periodic patterns for a model involving short-range and long-range interactions

The Impact of the Domain Boundary on an Inhibitory System: Existence and Location of a Stationary Half Disc

A stationary core-shell assembly in a ternary inhibitory system

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