A first-order calculus for the Monge-Ampere quasi-metric structure and its applications to Analysis and PDEs

Kansas State University, Manhattan KS

Investigators

Abstract

Our mathematical understanding of instantaneous rates of change stems from the invention of calculus by Newton and Leibniz in the 1680's and, by now, we have become accustomed to that notion in our daily lives. For instance, a quick look at our car's speedometer lets us know our speed at any given moment in time and a radar or GPS system can tell us how fast a storm is approaching or an airplane is flying at any given moment in time. There are, however, certain interesting mathematical contexts in which the notion of instantaneous speed cannot be taken for granted. That notion needs to be developed in an intrinsic fashion as to retain the natural properties and interpretations we are used to. Some of those properties can be expressed in the form of the so-called Sobolev or variational inequalities which relate position and speed at different scales. The pursue of natural notions of instantaneous speed in various mathematical contexts, as well as their variational inequalities and applications, stands as an increasingly active and exciting area of research in Analysis. Geometric and measure-theoretic objects can be associated to convex functions. The interplay between properties of these objects models the behavior of solutions to certain elliptic and parabolic PDEs as well as quasi-conformality of mappings with convex potentials, geometric features of some quasi-metric spaces, associated potential theory, etc. The research activities supported by this grant aim at developing such topics by means of a first-order calculus based on variational inequalities for the Monge-Ampere quasi-metric structure.