Micah Warren

Evans–Krylov Estimates for a nonconvex Monge–Ampère equation

We establish Evans–Krylov estimates for certain nonconvex fully nonlinear elliptic and parabolic equations by exploiting partial Legendre transformations. The equations under consideration arise in part from the study of the “pluriclosed flow” introduced by Streets and Tian (Int Math Res Not 16:3101–3133, 2010).

Nonpolynomial entire solutions to σk equations

ABSTRACT For 2k = n + 1, we exhibit nonpolynomial solutions to the Hessian equation on all of ℝn.

On solutions to Cournot–Nash equilibria equations on the sphere

We discuss equations associated to Cournot‐Nash Equilibria as put forward recently by Blanchet and Carlier. These equations are related to an optimal transport problem in which the source measure is known, but the target measure is part of the problem. The resulting equation is of Monge‐Ampere type with possible nonlocal terms. If the cost function is of a particular form, the equation is vulnerable to standard optimal transportation PDE techniques, with some modifications to deal with the new terms. We give some sufficient conditions for the problem on the sphere from which we can conclude that solutions are smooth.