### Abstract

^{2}(x, y), where d(·, ·) is the Riemannian distance of the round sphere; the second corresponds to the cost function -log |x − y|, known as the reflector antenna problem. We show that in both cases, the cost-sectional curvature is uniformly positive, and establish the geometrical properties so that the results of Loeper (in Acta Math., to appear) and Ma et al. (in Arch. Rational Mech. Anal. 177(2), 151-183 (2005)) can apply: global smooth solutions exist for arbitrary smooth positive data and optimal maps are Hölder continuous under weak assumptions on the data.

Original language | English |
---|---|

Pages (from-to) | 269-289 |

Number of pages | 21 |

Journal | Archive for Rational Mechanics and Analysis |

Volume | 199 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2011 |

Externally published | Yes |

### Cite this

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**Regularity of optimal maps on the sphere: the quadratic cost and the reflector antenna.** / Loeper, Gregoire.

Research output: Contribution to journal › Article › Research › peer-review

TY - JOUR

T1 - Regularity of optimal maps on the sphere: the quadratic cost and the reflector antenna

AU - Loeper, Gregoire

PY - 2011

Y1 - 2011

N2 - Building on the results of Ma et al. (in Arch. Rational Mech. Anal. 177(2), 151-183 (2005)), and of the author Loeper (in Acta Math., to appear), we study two problems of optimal transportation on the sphere: the first corresponds to the cost function d2(x, y), where d(·, ·) is the Riemannian distance of the round sphere; the second corresponds to the cost function -log |x − y|, known as the reflector antenna problem. We show that in both cases, the cost-sectional curvature is uniformly positive, and establish the geometrical properties so that the results of Loeper (in Acta Math., to appear) and Ma et al. (in Arch. Rational Mech. Anal. 177(2), 151-183 (2005)) can apply: global smooth solutions exist for arbitrary smooth positive data and optimal maps are Hölder continuous under weak assumptions on the data.

AB - Building on the results of Ma et al. (in Arch. Rational Mech. Anal. 177(2), 151-183 (2005)), and of the author Loeper (in Acta Math., to appear), we study two problems of optimal transportation on the sphere: the first corresponds to the cost function d2(x, y), where d(·, ·) is the Riemannian distance of the round sphere; the second corresponds to the cost function -log |x − y|, known as the reflector antenna problem. We show that in both cases, the cost-sectional curvature is uniformly positive, and establish the geometrical properties so that the results of Loeper (in Acta Math., to appear) and Ma et al. (in Arch. Rational Mech. Anal. 177(2), 151-183 (2005)) can apply: global smooth solutions exist for arbitrary smooth positive data and optimal maps are Hölder continuous under weak assumptions on the data.

UR - http://link.springer.com/content/pdf/10.1007%2Fs00205-010-0330-x.pdf

U2 - 10.1007/s00205-010-0330-x

DO - 10.1007/s00205-010-0330-x

M3 - Article

VL - 199

SP - 269

EP - 289

JO - Archive for Rational Mechanics and Analysis

JF - Archive for Rational Mechanics and Analysis

SN - 0003-9527

IS - 1

ER -