Partial Differential Equations With and Without Convexity Constraints

这个项目研究了非线性偏微分方程(PDE)中选定问题的精细定量性质，以及在有和没有称为凸性的结构条件下的变分。这些问题在数学的几个领域都有联系和应用，如分析、偏微分方程组、变分法和数值方法。此外，它们与其他科学和工程领域的重要应用同样相关。例如，本项目中研究的具有凸性约束的偏微分方程组和变分问题出现在不同的科学学科中，如物理学中的牛顿最小阻力问题、经济学中的垄断者问题(其中垄断者需要连同价格时间表一起设计产品线以最大化总利润)，或者弹性力学中的弹性壳的褶皱模式。尽管它无处不在，但带凸性约束的变分演算仍然鲜为人知。这个项目开发了新的数学工具，特别是偏微分方程技术，以更好地理解这一领域的基本问题。这个项目的一个重要主题是通过从其他领域引进思想和发展方法论来研究偏微分方程组和变分法中的基本问题，例如使用复杂的几何洞察力来解决经济学和物理学中出现的偏微分方程组问题，或者使用凸性技术来理解没有任何凸性的完全非线性偏微分方程组的精细性质。这个项目为研究生提供了培训机会，其结果将通过发表研究论文和讲座讲稿以及在国内和国际上的演讲向不同的受众传播。本项目集中于几类完全非线性椭圆型偏微分方程解的可解性、正则性估计和渐近分析，以及在有和没有凸性约束的变分中的问题，并将它们应用于分析和偏微分方程中的几个有趣的问题以及经济学和弹性力学中出现的问题。该项目由四个主要部分组成。第一个问题是研究在多大程度上可以用复杂几何中出现的奇异Abreu方程的解来逼近具有凸性约束的凸泛函的极小点(及其欧拉-拉格朗日方程)。其中一个泛函是经济学中垄断问题的罗切特-乔恩模型。第二部分建立了高度奇异的Abreu方程的整体可解性。这些四阶方程可以改写为Monge-Ampere方程组和线性化的Monge-Ampere方程组。第三部分研究了k-Hesse特征值问题的逆迭代格式的收敛问题。第四部分研究了完全非线性无凸椭圆型方程和线性化的Monge-Ampere方程的二阶导数估计的小可积指数相对于椭圆率的急剧衰减。首席研究员(PI)的目标是系统地发展Monge-Ampere型方程技术来研究没有任何凸性的完全非线性偏微分方程组的精细性质。此外，PI和他的合作者最近引入的PDE方法(如k-Hessian方程的分部非线性积分和Monge-Ampere类型的四阶方程的部分勒让德变换)将被进一步探索，以成功地解决作为该项目的一部分而要研究的问题。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.