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.
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
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.
- Some elementary examples of existence, nonexistence, uniqueness, and
nonuniqueness of optimal maps
- Optimal transport plans and the method of Kantorovich
- One dimensional problems
- The ODE version of the optimal transport problem
- The PDE version of the optimal transport problem
- Existence of optimal transport maps
- Mass optimization problems
MWF 3PM in Herman Brown 427
- Robert Hardt Office: Herman Brown 430; Office hours:
11-12 MWF (and others by appt.),
- Email: email@example.com, 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),
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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