The Monge-Ampère equation and its geometric applications

Quasi-neutral limit of the Euler-Poisson and Euler-Monge-Ampère systems

This paper studies the pressureless Euler-Poisson system and its fully non-linear counterpart, the Euler-Monge-Ampère system, where the fully non-linear Monge-Ampère equation substitutes for the linear Poisson equation. While the first is a model of plasma physics, the second is derived as a geometric approximation to the Euler incompressible equations. Using energy estimates, convergence of bo...

Geometric Properties of the Sections of Solutions to the Monge-ampère Equation

In this paper we establish several geometric properties of the cross sections of generalized solutions φ to the Monge-Ampère equation detD2φ = μ, when the measure μ satisfies a doubling property. A main result is a characterization of the doubling measures μ in terms of a geometric property of the cross sections of φ. This is used to obtain estimates of the shape and invariance properties of th...

Team Member: Bella Gu School: International School of Boston City: Cambridge, MA Supervisor: David Xianfeng Gu Title: Geometric Compression Based on Ricci Flow and Monge-Ampère Equation

Harnack’s Inequality for Solutions to the Linearized Monge-ampère Equation under Minimal Geometric Assumptions

We prove a Harnack inequality for solutions to LAu = 0 where the elliptic matrix A is adapted to a convex function satisfying minimal geometric conditions. An application to Sobolev inequalities is included.

The Monge-ampère Equation and Its Link to Optimal Transportation

We survey old and new regularity theory for the Monge-Ampère equation, show its connection to optimal transportation, and describe the regularity properties of a general class of Monge-Ampère type equations arising in that context.