On the global minimizers of a nonlocal isoperimetric problem in two dimensions

Sternberg, Peter; Topaloglu, Ihsan

doi:10.4171/ifb/252

Cited by 44 publications

(42 citation statements)

References 30 publications

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“…Sternberg and Topaloglu [66] have recently considered (NLIP) in the regime of small γ. They prove regularity of the phase boundary (for local minimizers) and prove that there is an interval of values for the mass constraint such that the global minimizer is exactly lamellar.…”

Section: Some Other Results For Global Minimizersmentioning

confidence: 99%

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

Choksi

2012

Quart. Appl. Math.

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Abstract. Energy-driven pattern formation induced by competing short and longrange interactions is common in many physical systems. In these proceedings we report on certain rigorous asymptotic results concerning global minimizers of a nonlocal perturbation to the well-known Ginzburg-Landau/Cahn-Hilliard free energy. We also discuss two hybrid numerical methods for accessing the ground states of these functionals.

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Section: Some Other Results For Global Minimizersmentioning

confidence: 99%

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

Choksi

2012

Quart. Appl. Math.

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“…Many solutions in two and three dimensions have been found that match the morphological phases in diblock copolymers [24,30,29,31,32,15,16,33,35,39]. Global minimizers of J B are studied in [2,37,19,5,18,17,11] for various parameter ranges. Applications of the second variation of J B and its connections to minimality and Gamma-convergence are found in [7,1,14].…”

Section: Introductionmentioning

confidence: 97%

Asymmetric and Symmetric Double Bubbles in a Ternary Inhibitory System

Ren¹,

Wei²

2014

SIAM J. Math. Anal.

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“…It is not clear for most values of ω and γ whether a global minimizer of J can have its interface meeting the domain boundary. To avoid this problem some authors assume that D is a rectangle and pose the Poisson equation (1.3) with the periodic boundary condition instead of the more natural Neumann boundary condition [5,10,11,15,26].…”

Section: Introductionmentioning

confidence: 99%

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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On the global minimizers of a nonlocal isoperimetric problem in two dimensions

Cited by 44 publications

References 30 publications

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

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

Asymmetric and Symmetric Double Bubbles in a Ternary Inhibitory System

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

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