Sobolev regularity for monge-ampère type equations

Guido De Philippis, Alessio Figalli

Research output: Contribution to journalArticlepeer-review


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 languageEnglish (US)
Pages (from-to)1812-1824
Number of pages13
JournalSIAM Journal on Mathematical Analysis
Issue number3
StatePublished - 2013


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

ASJC Scopus subject areas

  • Analysis
  • Computational Mathematics
  • Applied Mathematics


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