### Abstract

Language | English |
---|---|

Pages | 136-158 |

Number of pages | 23 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 72 |

Issue number | 1 |

Early online date | 17 Jul 2018 |

DOIs | |

State | Published - Jan 2019 |

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### Cite this

*Communications on Pure and Applied Mathematics*,

*72*(1), 136-158. DOI: 10.1002/cpa.21762

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*Communications on Pure and Applied Mathematics*, vol. 72, no. 1, pp. 136-158. DOI: 10.1002/cpa.21762

**The Equilibrium Measure for a Nonlocal Dislocation Energy.** / Mora, Maria Giovanna; Rondi, Luca; Scardia, Lucia.

Research output: Contribution to journal › Article

TY - JOUR

T1 - The Equilibrium Measure for a Nonlocal Dislocation Energy

AU - Mora,Maria Giovanna

AU - Rondi,Luca

AU - Scardia,Lucia

PY - 2019/1

Y1 - 2019/1

N2 - In this paper we characterize the equilibrium measure for a nonlocal and anisotropic weighted energy describing the interaction of positive dislocations in the plane. We prove that the minimum value of the energy is attained by a measure supported on the vertical axis and distributed according to the semicircle law, a well‐known measure that also arises as the minimizer of purely logarithmic interactions in one dimension. In this way we give a positive answer to the conjecture that positive dislocations tend to form vertical walls. This result is one of the few examples where the minimizer of a nonlocal energy is explicitly computed and the only one in the case of anisotropic kernels.

AB - In this paper we characterize the equilibrium measure for a nonlocal and anisotropic weighted energy describing the interaction of positive dislocations in the plane. We prove that the minimum value of the energy is attained by a measure supported on the vertical axis and distributed according to the semicircle law, a well‐known measure that also arises as the minimizer of purely logarithmic interactions in one dimension. In this way we give a positive answer to the conjecture that positive dislocations tend to form vertical walls. This result is one of the few examples where the minimizer of a nonlocal energy is explicitly computed and the only one in the case of anisotropic kernels.

U2 - 10.1002/cpa.21762

DO - 10.1002/cpa.21762

M3 - Article

VL - 72

SP - 136

EP - 158

JO - Communications on Pure and Applied Mathematics

T2 - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 1

ER -