### Abstract

*C*

^{1}regularity of c-convex weak Alexandrov solutions of a Monge-Ampère type equation in dimension two, assuming only a bound from above on the Monge-Ampère measure. The Monge-Ampère equation involved arises in the optimal transport problem. Our result holds true under a natural condition on the cost function, namely

*non-negative cost-sectional curvature*, a condition introduced in Ma et al. (Arch Ration Mech Anal 177(2):151-183, 2005), that was shown in Loeper (Acta Math, to appear) to be necessary for

*C*

^{1}regularity. Such a condition holds in particular for the case "cost = distance squared" which leads to the usual Monge-Ampère equation det

*D*

^{2}

*u = f*. Our result is in some sense optimal, both for the assumptions on the density [thanks to the regularity counterexamples of Wang (Proc Am Math Soc 123(3):841-845, 1995)] and for the assumptions on the cost-function [thanks to the results of Loeper (Acta Math, to appear)].

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

Pages (from-to) | 537-550 |

Number of pages | 14 |

Journal | Calculus of Variations and Partial Differential Equations |

Volume | 35 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2009 |

Externally published | Yes |

### Cite this

}

*C*

^{1}regularity of solutions of the Monge-Ampere equation for optimal transport in dimension two',

*Calculus of Variations and Partial Differential Equations*, vol. 35, no. 4, pp. 537-550. https://doi.org/10.1007/s00526-009-0222-9

** C^{1} regularity of solutions of the Monge-Ampere equation for optimal transport in dimension two.** / Figalli, Alessio; Loeper, Gregoire.

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

TY - JOUR

T1 - C1 regularity of solutions of the Monge-Ampere equation for optimal transport in dimension two

AU - Figalli, Alessio

AU - Loeper, Gregoire

PY - 2009

Y1 - 2009

N2 - We prove C1 regularity of c-convex weak Alexandrov solutions of a Monge-Ampère type equation in dimension two, assuming only a bound from above on the Monge-Ampère measure. The Monge-Ampère equation involved arises in the optimal transport problem. Our result holds true under a natural condition on the cost function, namely non-negative cost-sectional curvature, a condition introduced in Ma et al. (Arch Ration Mech Anal 177(2):151-183, 2005), that was shown in Loeper (Acta Math, to appear) to be necessary for C1 regularity. Such a condition holds in particular for the case "cost = distance squared" which leads to the usual Monge-Ampère equation det D2u = f. Our result is in some sense optimal, both for the assumptions on the density [thanks to the regularity counterexamples of Wang (Proc Am Math Soc 123(3):841-845, 1995)] and for the assumptions on the cost-function [thanks to the results of Loeper (Acta Math, to appear)].

AB - We prove C1 regularity of c-convex weak Alexandrov solutions of a Monge-Ampère type equation in dimension two, assuming only a bound from above on the Monge-Ampère measure. The Monge-Ampère equation involved arises in the optimal transport problem. Our result holds true under a natural condition on the cost function, namely non-negative cost-sectional curvature, a condition introduced in Ma et al. (Arch Ration Mech Anal 177(2):151-183, 2005), that was shown in Loeper (Acta Math, to appear) to be necessary for C1 regularity. Such a condition holds in particular for the case "cost = distance squared" which leads to the usual Monge-Ampère equation det D2u = f. Our result is in some sense optimal, both for the assumptions on the density [thanks to the regularity counterexamples of Wang (Proc Am Math Soc 123(3):841-845, 1995)] and for the assumptions on the cost-function [thanks to the results of Loeper (Acta Math, to appear)].

UR - http://link.springer.com/content/pdf/10.1007%2Fs00526-009-0222-9.pdf

U2 - 10.1007/s00526-009-0222-9

DO - 10.1007/s00526-009-0222-9

M3 - Article

VL - 35

SP - 537

EP - 550

JO - Calculus of Variations and Partial Differential Equations

JF - Calculus of Variations and Partial Differential Equations

SN - 0944-2669

IS - 4

ER -