Quasiconvex Functions and Nonlinear PDE's

这个项目是对拟凸函数的深入分析(也称为。水平集凸函数)，包括非光滑拟凸函数。它侧重于它们与二阶、完全非线性偏微分方程的联系，以及这些联系在各种问题中的应用，如最坏情况设计、随机目标、曲率运动和最优运输。其基本思想是利用偏微分方程粘性解的理论来刻画函数何时具有凸水平集。从这一点开始，接下来将进行几个分支的调查。例如，通过一个非线性偏微分方程刻画拟凸函数，可以得到给定函数的拟凸包络的偏微分方程解。类似于已知的给定函数的凸包络的偏微分方程组的构造。另一种可能的构造是多元函数的拟凸-拟凹包络(这在连续博弈论中有重要的应用)。有趣的是，在用偏微分方程刻画拟凸函数和用平均曲率刻画由边界运动演变的曲面之间有很强的联系。这个项目可以产生一种更稳定的曲率近似运动类型。本项目还将研究拟凸特征微分方程、随机目标问题和确定性微分拔河游戏之间的联系。最后，拟凸函数在变分演算中自然出现，代价是最坏的。这个项目将考虑几个这类问题，包括具有最坏转移成本的最优运输理论的扩展，以及具有各种约束的变分问题。水平集凸性)是凸性的一个重要推广。拟凸函数在经济学和管理学中被广泛使用了几十年，它们自然而然地作为偏好/目标函数出现。它在最优控制和具有最坏情况代价的微分对策中有很重要的应用和推广。在结构工程、材料科学、图像处理、半导体设计、飞机着陆策略和癌症管理的化疗剂量方面的重要应用都属于最坏情况设计的范畴，涉及本项目所考虑的类型的功能。由边界物理控制的空间区域的扩展，例如火灾等离子体的边缘(例如，森林大火的前缘)，由本项目中推导和研究的方程决定。与随机目标问题的联系是控制问题的一种重要方法，在这些问题中，人们寻求以比简单的平均目标值更确定的方式达到目标。这类问题的应用范围从进化生物学到图像重建。在经典的最优传输中，人们试图最小化将一种分布的粒子传输到另一种分布的粒子的平均成本，同时保持总质量。这个项目将研究将颗粒运输的最大成本降至最低。这些问题在分子重排的物理学、公共交通网络的建设以及恶性肿瘤如何转移的生物化学中都有应用。

This project is an in depth analysis of quasiconvex functions (a.k.a. level set convex functions), including nonsmooth quasiconvex functions. It focuses on their connections with second order, fully nonlinear partial differential equations, and the application of these connections to diverse problems such as worst case design, stochastic targeting, motion by curvature, and optimal transport. The primary idea is to use the theory of viscosity solutions of partial differential equations to characterize when a function has convex level sets. From this point several branches of inquiry will be followed. For example, the characterization of quasiconvex functions through a nonlinear partial differential equation suggests a PDE construction of the quasiconvex envelope of a given function ? analogous to the well known PDE construction of the convex envelope of a given function. Another possible construction is the quasiconvex-quasiconcave envelope of a function of several variables (which would have significant application in continuous game theory). Interestingly, there is a very strong connection between the characterization of quasiconvex functions using partial differential equations and the characterization of surfaces evolving by the motion of the boundary by mean curvature. This project could yield a more stable type of approximate motion by curvature. This project will also study of the connection between the quasiconvex characterizing differential equation, stochastic target problems, and deterministic differential tug-of-war games. Finally, quasiconvex functions arise naturally in the calculus of variations with a worst case cost. This project will consider several of problems of this type including an extension to optimal transport theory with worst case cost of transfer, and variational problems with various constraints.Quasiconvexity (a.k.a. level set convexity) is an important generalization of convexity. Quasiconvex functions have been widely used for decades in economics and management science, where they arise naturally as preference/objective functions. A very important application and extension was found in optimal control and differential games with a worst case cost. Significant applications to structural engineering, materials science, image processing, semiconductor design, aircraft landing strategies, and chemotherapy dosing for carcinoma management all fall in the category of worst case design and involve functions of the type considered in this project. Expansion of a region in space governed by the physics at the boundary such as the edge of conflagration plasma (e.g., the front of a forest fire) is determined by equations which are derived and studied in this project. The connection with stochastic target problems is an important approach to control problems in which one seeks to reach an objective with much more certainty than simply the average objective value. Such problems have applications ranging from evolutionary biology to image reconstruction. In classical optimal transport one tries to minimize the average cost of transporting one distribution of particles to another while maintaining the total mass. This project will study minimizing the maximum cost of transportation of the particles. Such problems have applications in the physics of the rearrangement of molecules, the construction of public transportation networks, and in the biochemistry of how malignant tumors metastasize.