Diego Maldonado
Kansas State University

A First-Order Calculus Associated to the Monge-Ampere Equation

Abstract
Geometric and measure-theoretic objects can be associated to certain convex functions in $\mathbb{R}^n$. These objects include a quasi-distance and a Borel measure in $\mathbb{R}^n$ which render a structure of space of homogeneous type. We will describe how real-analysis techniques in this quasi-metric space, complemented by the existence of related Sobolev and Poincare inequalities, can be applied to the regularity theory of convex solutions $u$ to the Monge-Ampere equation $\det D^2u=f$ as well as solutions $v$ of the linearized Monge-Ampere equation $L(v)=g$.
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