# Exact Periodic Stripes for Minimizers of a Local/Nonlocal Interaction Functional in General Dimension

@article{Daneri2018ExactPS, title={Exact Periodic Stripes for Minimizers of a Local/Nonlocal Interaction Functional in General Dimension}, author={Sara Daneri and Eris Runa}, journal={Archive for Rational Mechanics and Analysis}, year={2018}, volume={231}, pages={519-589} }

We study the functional considered in Giuliani et al. (Phys Rev B 84:064205, 2011, Commun Math Phys 331(1):333–350, 2014) and Giuliani and Seiringer (Commun Math Phys 347:983–1007, 2016) and a continuous version of it, analogous to the one considered in Goldman and Runa (On the optimality of stripes in a variational model with nonlocal interactions, 2016. arXiv:1611.07228). The functionals consist of a perimeter term and a nonlocal term which are in competition. For both the continuous and…

## 23 Citations

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- Physics, Computer ScienceSIAM J. Math. Anal.
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In this paper we study pattern formation for a local/nonlocal interaction functional where the local attractive term is given by the $1$-perimeter and the nonlocal repulsive term is the Yukawa (or…

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The aim of this note is to review some recent results on a family of functionals penalizing oblique oscillations. These functionals naturally appeared in some varia-tional problem related to pattern…

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We study a geometric variational problem for sets in the plane in which the perimeter and a regularized dipolar interaction compete under a mass constraint. In contrast to previously studied nonlocal…

## References

SHOWING 1-10 OF 58 REFERENCES

On the optimality of stripes in a variational model with non-local interactions

- Mathematics, PhysicsCalculus of Variations and Partial Differential Equations
- 2019

We study pattern formation for a variational model displaying competition between a local term penalizing interfaces and a non-local term favoring oscillations. By means of a $$\Gamma $$Γ-convergence…

Uniform energy distribution for an isoperimetric problem with long-range interactions

- Mathematics
- 2008

We study minimizers of a nonlocal variational problem. The problem is a mathematical paradigm for the ubiquitous phenomenon of energy-driven pattern formation induced by competing short- and…

On an Isoperimetric Problem with a Competing Nonlocal Term II: The General Case

- Mathematics, Physics
- 2013

This paper is the continuation of a previous paper (H. Knupfer and C. B. Muratov, Comm. Pure Appl. Math. 66 (2013), 1129‐1162). We investigate the classical isoperimetric problem modified by an…

Formation of Stripes and Slabs Near the Ferromagnetic Transition

- Mathematics, Physics
- 2014

We consider Ising models in d = 2 and d = 3 dimensions with nearest neighbor ferromagnetic and long-range antiferromagnetic interactions, the latter decaying as (distance)−p, p > 2d, at large…

The Γ-Limit of the Two-Dimensional Ohta–Kawasaki Energy. I. Droplet Density

- Mathematics, Physics
- 2013

This is the first in a series of two papers in which we derive a Γ-expansion for a two-dimensional non-local Ginzburg–Landau energy with Coulomb repulsion, also known as the Ohta–Kawasaki model, in…

Periodic Striped Ground States in Ising Models with Competing Interactions

- Physics, Mathematics
- 2015

We consider Ising models in two and three dimensions, with short range ferromagnetic and long range, power-law decaying, antiferromagnetic interactions. We let J be the ratio between the strength of…

On an isoperimetric problem with a competing non-local term. I. The planar case

- Physics, Mathematics
- 2011

This paper is concerned with a study of the classical isoperimetric problem modified by an addition of a non-local repulsive term. We characterize existence, non-existence and radial symmetry of the…

Low Density Phases in a Uniformly Charged Liquid

- Physics, Mathematics
- 2016

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…

Small Volume Fraction Limit of the Diblock Copolymer Problem: I. Sharp-Interface Functional

- Mathematics, Computer ScienceSIAM J. Math. Anal.
- 2010

This article addresses the limit in which epsilon and the volume fraction tend to zero but the number of minority phases (called particles) remains O(1), and focuses on two levels of Gamma-convergence, which derive first- and second-order effective energies, whose energy landscapes are simpler and more transparent.

Checkerboards, stripes, and corner energies in spin models with competing interactions

- Physics, Mathematics
- 2011

We study the zero temperature phase diagram of Ising spin systems in two dimensions in the presence of competing interactions, long range antiferromagnetic and nearest neighbor ferromagnetic of…