Sobolev regularity for monge-ampère type equations

Guido De Philippis; Alessio Figalli

doi:10.1137/120898619

Sobolev regularity for monge-ampère type equations

Guido De Philippis, Alessio Figalli

Mathematics

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

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	https://doi.org/10.1137/120898619
State	Published - 2013

Keywords

A priori estimates
Higher integrability
Monge-Ampère equation
Sobolev regularity

ASJC Scopus subject areas

Analysis
Computational Mathematics
Applied Mathematics

Access to Document

10.1137/120898619

Cite this

@article{8727d55b54404466a221965815082c75,

title = "Sobolev regularity for monge-amp{\`e}re type equations",

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-Amp{\`e}re 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].",

keywords = "A priori estimates, Higher integrability, Monge-Amp{\`e}re equation, Sobolev regularity",

author = "{De Philippis}, Guido and Alessio Figalli",

year = "2013",

doi = "10.1137/120898619",

language = "English (US)",

volume = "45",

pages = "1812--1824",

journal = "SIAM Journal on Mathematical Analysis",

issn = "0036-1410",

publisher = "Society for Industrial and Applied Mathematics Publications",

number = "3",

}

TY - JOUR

T1 - Sobolev regularity for monge-ampère type equations

AU - De Philippis, Guido

AU - Figalli, Alessio

PY - 2013

Y1 - 2013

N2 - 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].

AB - 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].

KW - A priori estimates

KW - Higher integrability

KW - Monge-Ampère equation

KW - Sobolev regularity

UR - http://www.scopus.com/inward/record.url?scp=84884790464&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84884790464&partnerID=8YFLogxK

U2 - 10.1137/120898619

DO - 10.1137/120898619

M3 - Article

AN - SCOPUS:84884790464

SN - 0036-1410

VL - 45

SP - 1812

EP - 1824

JO - SIAM Journal on Mathematical Analysis

JF - SIAM Journal on Mathematical Analysis

IS - 3

ER -

Sobolev regularity for monge-ampère type equations

Abstract

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this