### Abstract

Original language | English |
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Pages (from-to) | 1-18 |

Journal | Communications in Mathematical Physics |

Early online date | 24 Apr 2019 |

DOIs | |

Publication status | E-pub ahead of print - 24 Apr 2019 |

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Communications in Mathematical Physics*, 1-18. https://doi.org/10.1007/s00220-019-03368-w

**The ellipse law: Kirchhoff meets dislocations.** / Carrillo, J.A.; Mateu, J; Mora, Maria Giovanna; Rondi, Luca; Scardia, Lucia; Verdera, J.

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, pp. 1-18. https://doi.org/10.1007/s00220-019-03368-w

}

TY - JOUR

T1 - The ellipse law: Kirchhoff meets dislocations

AU - Carrillo, J.A.

AU - Mateu, J

AU - Mora, Maria Giovanna

AU - Rondi, Luca

AU - Scardia, Lucia

AU - Verdera, J

PY - 2019/4/24

Y1 - 2019/4/24

N2 - In this paper we consider a nonlocal energy Iα whose kernel is obtained by adding to the Coulomb potential an anisotropic term weighted by a parameter α∈R . The case α = 0 corresponds to purely logarithmic interactions, minimised by the circle law; α = 1 corresponds to the energy of interacting dislocations, minimised by the semi-circle law. We show that for α∈(0,1) the minimiser is the normalised characteristic function of the domain enclosed by the ellipse of semi-axes 1−α−−−−−√ and 1+α−−−−−√ . This result is one of the very few examples where the minimiser of a nonlocal anisotropic energy is explicitly computed. For the proof we borrow techniques from fluid dynamics, in particular those related to Kirchhoff’s celebrated result that domains enclosed by ellipses are rotating vortex patches, called Kirchhoff ellipses.

AB - In this paper we consider a nonlocal energy Iα whose kernel is obtained by adding to the Coulomb potential an anisotropic term weighted by a parameter α∈R . The case α = 0 corresponds to purely logarithmic interactions, minimised by the circle law; α = 1 corresponds to the energy of interacting dislocations, minimised by the semi-circle law. We show that for α∈(0,1) the minimiser is the normalised characteristic function of the domain enclosed by the ellipse of semi-axes 1−α−−−−−√ and 1+α−−−−−√ . This result is one of the very few examples where the minimiser of a nonlocal anisotropic energy is explicitly computed. For the proof we borrow techniques from fluid dynamics, in particular those related to Kirchhoff’s celebrated result that domains enclosed by ellipses are rotating vortex patches, called Kirchhoff ellipses.

UR - http://www.scopus.com/inward/record.url?scp=85064830525&partnerID=8YFLogxK

U2 - 10.1007/s00220-019-03368-w

DO - 10.1007/s00220-019-03368-w

M3 - Article

SP - 1

EP - 18

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

ER -