Sobolev regularity for monge-ampère type equations

Guido De Philippis, Alessio Figalli

Research output: Contribution to journalArticle

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 languageEnglish (US)
Pages (from-to)1812-1824
Number of pages13
JournalSIAM Journal on Mathematical Analysis
Volume45
Issue number3
DOIs
StatePublished - 2013

Keywords

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

ASJC Scopus subject areas

  • Analysis
  • Computational Mathematics
  • Applied Mathematics

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