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

我们对瞬时变化率的数学理解源于16世纪80年代牛顿和莱布尼茨在S发明的微积分，到现在，我们已经在日常生活中习惯了这个概念。例如，快速查看我们的汽车速度表就能知道我们在任何给定时刻的速度，雷达或GPS系统可以告诉我们风暴正在以多快的速度逼近或飞机在任何给定时刻飞行。然而，在某些有趣的数学背景下，瞬时速度的概念不能被视为理所当然。这一概念需要以一种内在的方式发展，以保留我们习惯的自然属性和解释。其中一些性质可以用所谓的索博列夫或变分不等式的形式来表示，它们在不同的尺度上将位置和速度联系在一起。在不同的数学背景下追求瞬时速度的自然概念，以及它们的变分不等式和应用，是分析中一个日益活跃和令人兴奋的研究领域。几何和测度论对象可以与凸函数联系在一起。这些对象的性质之间的相互作用模拟了某些椭圆型和抛物型偏微分方程解的行为，以及具有凸势的映射的拟一致，一些拟度量空间的几何特征，结合势理论等。这项拨款支持的研究活动旨在通过基于Monge-Ampere拟度量结构的变分不等式的一阶微积分来发展这些主题。

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.