Archive for Rational Mechanics and Analysis

, Volume 212, Issue 2, pp 359–414 | Cite as

A Perturbation Argument for a Monge–Ampère Type Equation Arising in Optimal Transportation

  • Luis Caffarelli
  • María del Mar GonzálezEmail author
  • Truyen Nguyen


We prove some interior regularity results for potential functions of optimal transportation problems with power costs. The main point is that our problem is equivalent to a new optimal transportation problem whose cost function is a sufficiently small perturbation of the quadratic cost, but it does not satisfy the well known condition (A.3) guaranteeing regularity. The proof consists in a perturbation argument from the standard Monge–Ampère equation in order to obtain, first, interior C1,1 estimates for the potential and, second, interior Hölder estimates for second derivatives. In particular, we take a close look at the geometry of optimal transportation when the cost function is close to quadratic in order to understand how the equation degenerates near the boundary.


Convex Function Minimum Point Type Equation Comparison Principle Universal Constant 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Luis Caffarelli
    • 1
  • María del Mar González
    • 2
    Email author
  • Truyen Nguyen
    • 3
  1. 1.University of Texas at AustinAustinUSA
  2. 2.Universitat Politècnica de CatalunyaBarcelonaSpain
  3. 3.University of AkronAkronUSA

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