Given a distribution f of iron mines throughout the countryside, and
a distribution g of factories which require iron ore, the optimal
transportation problem of Monge and Kantorovich asks for the mines to
be paired with the factories so as to minimize the average (say)
Euclidean distance squared between mine and factory. This problem is
deeply connected to geometry, inequalities, and nonlinear differential
equations, with applications ranging from shape recognition to weather
prediction.
In the talk I discuss what happens when the production capacity of the
mines need not agree with the demand of the factories, so one ships
only a certain fraction of the ore being produced, again choosing the
locations of factories and mines which remain active so as to minimize
total transportation costs. If the mines are continuously distributed
in Euclidean space, and positively separated from the factories, the
solution will be unique, and is given by pair of domains U,V Ì Rn, with U containing the active mines and V the active
factories. These domains are characterized as the non-contact regions
in a double obstacle problem for the Monge-Ampère equation. We go
on to specify conditions on f and g which are sufficient to ensure
that U and V have continuously differentiable free boundaries, and
that the correspondence s :[`(U)] ®[`(V)] mapping mines to factories is a homeomorphism or smoother,
Hölder continuous up to the free (and part of the fixed)
boundary.
