Diego Maldonado
Kansas State University
A FirstOrder Calculus Associated to the MongeAmpere Equation
 Abstract

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