The Monopolist's Problem: Optimization over the space of convex functions and a new free boundary problem

In this talk I discuss joint work with Robert McCann and Kelvin Shuangjian Zhang concerning the Monopolists’ problem. This problem comes from a simple economics model which displays rich mathematical behaviour and lies at the intersection of optimal transport, free boundary problems, and convex analysis. Mathematically one aims to minimize a uniformly convex Lagrangian, however restricted to the space of convex functions. The requirement that the minimization take place over the space of convex functions leads to a free boundary between the regions of strict and nonstrict convexity and in these regions the solution displays qualitatively different behaviour. In this talk I outline recent work in which we prove results on the configuration of the different domains, regularity of the free boundary, and completely describe the solution in a prototypical case.