### Abstract

In this note we prove that if the cost function satisfies some necessary structural conditions and the densities are bounded away from zero and infinity, then strictly c-convex potentials arising in optimal transportation belong to W2,1+κ loc for some κ > 0. This generalizes some recents results concerning the regularity of strictly convex Alexandrov solutions of the Monge-Ampère equation with right-hand side bounded away from zero and infinity [G. De Philippis and A. Figalli, Invent. Math., 192 (2013), pp. 55-69; G. De Philippis, A. Figalli, and O. Savin, Math. Ann., to appear; T. Schmidt, Adv. Math., to appear].

Original language | English (US) |
---|---|

Pages (from-to) | 1812-1824 |

Number of pages | 13 |

Journal | SIAM Journal on Mathematical Analysis |

Volume | 45 |

Issue number | 3 |

DOIs | |

State | Published - 2013 |

### Keywords

- A priori estimates
- Higher integrability
- Monge-Ampère equation
- Sobolev regularity

### ASJC Scopus subject areas

- Analysis
- Computational Mathematics
- Applied Mathematics

## Fingerprint Dive into the research topics of 'Sobolev regularity for monge-ampère type equations'. Together they form a unique fingerprint.

## Cite this

De Philippis, G., & Figalli, A. (2013). Sobolev regularity for monge-ampère type equations.

*SIAM Journal on Mathematical Analysis*,*45*(3), 1812-1824. https://doi.org/10.1137/120898619