Zoom link: https://princeton.zoom.us/j/594605776

The optimal transport problem, formulated by Gaspard Monge in 1781, asks whether or not it is possible to find a map minimizing the total cost of moving a distribution of mass $f$ to another $g$ given that the cost of moving from $x$ to $y$ is measured by $c = c(x,y)$. The most fundamental case is that of the quadratic cost on $\R^n$, when $c(x,y) = |x-y|^2$. In this setting, this problem is a generalization of the second boundary value problem for the Monge--Amp\`{e}re equation. In this talk, we will discuss a study of the most common image and informal description of the optimal transport problem, what is the most efficient way to fill a hole with a given pile of sand? More precisely, we discuss some regularity results for the optimal transport problem, for quadratic cost, which include the case that $f$ and $g$ are absolutely continuous measures concentrated on bounded convex domains with densities that behave like positive powers of the distance functions to the boundaries of these domains.

This is joint work with Ovidiu Savin.