Regularity of the Monge–Ampère equation in Besov’s spaces

  • Alexander V. Kolesnikov
  • Sergey Yu. TikhonovEmail author


Let \(\mu = e^{-V} \ dx\) be a probability measure and \(T = \nabla \Phi \) be the optimal transportation mapping pushing forward \(\mu \) onto a log-concave compactly supported measure \(\nu = e^{-W} \ dx\). In this paper, we introduce a new approach to the regularity problem for the corresponding Monge–Ampère equation \(e^{-V} = \det D^2 \Phi \cdot e^{-W(\nabla \Phi )}\) in the Besov spaces \(W^{\gamma ,1}_{loc}\). We prove that \(D^2 \Phi \in W^{\gamma ,1}_{loc}\) provided \(e^{-V}\) belongs to a proper Besov class and \(W\) is convex. In particular, \(D^2 \Phi \in L^p_{loc}\) for some \(p>1\). Our proof does not rely on the previously known regularity results.

Mathematics Subject Classification (2000)

Primary 35J60 35B65 Secondary 46E35 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Alexander V. Kolesnikov
    • 1
  • Sergey Yu. Tikhonov
    • 2
    Email author
  1. 1.Higher School of EconomicsMoscowRussia
  2. 2.ICREA and Centre de Recerca MatemàticaBellaterraSpain

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