MATH 521: Topics in Real Analysis, Fall 2001

Optimal Transport Problems


Recent years have seen much new mathematical literature on some old problems credited to Monge (1781) and Kantorovitch (1942) concerning the optimal transportation of mass.  The original formulation is:  Given two distributions with equal masses of a given material described by functions  f(x)  and  g(x)  (corresponding for example to an embankment or pile of dirt and an escavation or hole), find a transport map  y  which carries the first distribution into the second and minimization the transportation cost.

    C(y)  =  J|x-y| f(x) dx .

More generally one can replace the functions  f , g  by positive measures of equal mass and replace the simple distance  |x-y|  by a more general cost function  c(x,y)  and again seek an optimal transportation map  y(x) .  The problem is now seen to touch on many fields, including fluid mechanics, partial differential equations, shape optimization, and probabliity theory.  Fortunately, there are now several lecture notes and papers providing an introduction to the new and old results.  We will start by following those of L. Ambrosio which focus more on the case  c(x,y) = |x-y| , the geometric aspects of the problems, and some of the most accessible recent results of Brenier, Buttazzo, Caffarelli, Evans, Feldman, Gangbo, McCann, and Sudakov.  A brief outline of these notes includes the following:

We will also work through some of Evans"s two lecture notes which emphasize more the PDE aspects.  Prerequisites for the course include some knowledge of elementary measure theore as in Math 425.


MWF 3PM in Herman Brown 427


Robert Hardt    Office: Herman Brown 430; Office hours: 11-12 MWF (and others by appt.),
    Email:, Telephone: ext 3280



L. Ambrosio, Lecture Notes on Optimal Transport Problems.  Scuola Normale Superiore, Pisa, 2000.

L.C. Evans, Partial Differential Equations and the Monge-Kantorovich Mass Transfer in Current Developments in Mathematics, R. Bott et al. ed. International Press, Cambridge, 1997.

L.C. Evans and W. Gangbo,  Differential Equation Methods for the Monge-Kantorovich Mass Transfer Problem. Memoirs of the A.M.S. 137(1999), 1-66.

Consumer Warning:

Much of this material is new to the instructor, and the active help ofanyone attending will be greatly appreciated.


Proof that a diffuse measure takes on all possible values.

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