## Abstract

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.

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

Publication status | Published - 18 Nov 2018 |

## ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics