Wilfrid Gangbo
Mass transport problems, its Connections and Applications

ABSTRACT. These lectures are introduction to the Monge-Kantorovich Theory (MKT). The basic problem, which occurs naturally in economics, can be formulated as: Given two probability measures on ${\bf R}^d$, representing the distribution of production $\mu$ and consumption $\nu$ for some commodity, the problem is to determine the most efficient way to rearrange the mass of the first distribution to yield the second. Efficiency is measured against a function $c(x,y) \geq 0$ which specifies the cost per unit mass for transporting mass from $x \in {\bf R}^d$ to $y \in {\bf R}^d$. We develope the basic theory of optimal maps and state fundamental results recently obtained by various authors. Applications we plan to cover are dynamical evolution problems which appear in fluids mechanic and kinetic theory. As time permits, we will mention applications to shape optimization.