Partial Differential Equations With and Without Convexity Constraints

本项目研究非线性偏微分方程（PDE）中选定问题的精细定量性质以及具有和不具有称为凸性的结构条件的变分法。这些问题在数学的几个领域都有联系和应用，如分析、偏微分方程、变分法和数值方法。此外，它们同样与其他科学和工程领域的重要应用有关。例如，本课题中研究的具有凸性约束的偏微分方程和变分法问题，在物理学中的牛顿最小阻力问题、经济学中的垄断者问题（垄断者为了使总利润最大化，需要设计生产线和价格表）、弹性学中的弹性壳模型等不同的科学领域中产生。尽管它的普遍性，凸约束的变分法仍然知之甚少。该项目开发新的数学工具，特别是PDE技术，以更好地理解这一领域的基本问题。该项目的一个重要主题是通过从其他领域引入思想和开发方法来研究PDE和变分法的一个领域的基本问题，例如使用复杂的几何见解来解决经济学和物理学中出现的PDE问题，或者使用凸性技术来理解完全非线性PDE的精细特性。本研究计画为研究生提供训练机会，其成果将透过发表研究论文、演讲稿，以及在国内、国际演讲会等方式，传播给不同的听众。本研究计画的重点是几类完全非线性椭圆型偏微分方程与变分法问题的可解性、正则性估计与渐近分析，有和没有凸约束，并将其应用于分析和PDE中的几个有趣的问题，以及经济学和弹性中出现的问题。该项目包括四个主要部分。第一个调查在何种程度上可以近似极小（和他们的欧拉-拉格朗日方程）的凸性约束的凸泛函的奇异Abreu方程的解决方案，出现在复杂的几何形状。其中一个泛函是经济学中垄断者问题的罗切特-乔内模型。第二部分研究了高奇异Abreu方程的整体可解性。这些四阶方程可以重写为Monge-Ampere方程和线性化Monge-Ampere方程的系统。第三部分研究了k-Hessian特征值问题的逆迭代格式的收敛性。最后一部分研究了无凸性的完全非线性椭圆方程和线性化的Monge-Ampere方程的二阶导数估计中小可积指数随椭圆率的急剧衰减。主要研究者（PI）的目标是系统地发展Monge-Ampere型方程技术，以研究没有任何凸性的完全非线性偏微分方程的精细性质。此外，委员会认为，PI及其合作者介绍的最新PDE方法（如k-Hessian方程的非线性分部积分和Monge-Ampere型四阶方程的部分Legendre变换）该奖项反映了NSF的法定使命，并通过使用基金会的知识产权进行评估，被认为值得支持。优点和更广泛的影响审查标准。

This project studies fine quantitative properties of selected problems in nonlinear partial differential equations (PDE) and the calculus of variations with and without a structural condition called convexity. These problems have connections and applications in several areas of mathematics such as analysis, PDEs, the calculus of variations, and numerical methods. Moreover, they are equally relevant to important applications in other areas of science and engineering. For example, the PDE and calculus of variations problems with a convexity constraint investigated in this project arise in different scientific disciplines such as Newton’s problem of minimal resistance in physics, the monopolist’s problem (where the monopolist needs to design the product line together with a price schedule so as to maximize the total profit) in economics, or wrinkling patterns of elastic shells in elasticity. Despite its ubiquity, the calculus of variations with a convexity constraint is still poorly understood. This project develops new mathematical tools, especially PDE techniques, to better understand basic problems in this area. An important theme of this project is to investigate fundamental questions in one field of PDE and the calculus of variations by importing ideas and developing methodologies from other fields, such as using complex geometric insights to tackle PDE questions arising in economics and physics, or using convexity techniques to understand fine properties of fully nonlinear PDEs without any convexity. This project provides training opportunities for graduate students, and its results will be disseminated to diverse audiences via publications of research papers and lecture notes and via presentations at national and international venues.This project focuses on the solvability, regularity estimates, and asymptotic analysis, of several classes of fully nonlinear elliptic PDE and problems in the calculus of variations, with and without convexity constraints and apply them to several interesting problems in analysis and PDE and those arising in economics and elasticity. The project consists of four main parts. The first one investigates to what extent one can approximate the minimizers (and their Euler-Lagrange equations) of convex functionals with a convexity constraint by solutions of singular Abreu equations which arise in complex geometry. One of such functionals is the Rochet-Chone model for the monopolist's problem in economics. The second part aims to establish the global solvability of highly singular Abreu equations. These fourth order equations can be rewritten as systems of a Monge-Ampere equation and a linearized Monge-Ampere equation. The third part studies the convergence of an inverse iterative scheme for the k-Hessian eigenvalue problem. The last part investigates the sharp decay, with respect to the ellipticity ratio, for the small integrability exponent in the second derivative estimates for fully nonlinear elliptic equations without convexity, and linearized Monge-Ampere equations. The principal investigator (PI) aims to systematically develop the Monge-Ampere type equation techniques to study fine properties of fully nonlinear PDE without any convexity. Moreover, recent PDE methods introduced by the PI and his collaborators (such as nonlinear integration by parts for k-Hessian equations and partial Legendre transforms for fourth order equations of Monge-Ampere type) will be further explored to successfully attack the problems to be investigated as part of this project.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.