# What is the most efficient way to fill a hole with a given pile of sand?

-
Yash Jhaveri, Columbia University

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.