Problems related to the infinity Laplacian operator, the weak KAM theory and singularities of solutions of Monge-Ampere equations

无穷大拉普拉斯算子、弱KAM理论和Monge-Ampere方程解的奇点相关问题

• 批准号: 0901460
• 负责人: Yifeng Yu
• 金额: $ 33.29万
• 依托单位: University of California-Irvine
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-06-01 至 2013-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901460&HistoricalAwards=false
• 关键词: Problems; related; infinity; Laplacian; operator

YuThe PI proposes to continue his study of problems related to the infinity Laplacian operator, the weak KAM theory and singularities of solutions of the Monge-Ampere equation. (1) The infinity Laplacian operator arises from minimizing the L-infinity norm of the gradient and a two person differential game called ?tug-of-war?. The PI intends to solve several problems from the?tug-of-war? game. One of the important questions is to see how we can use the game theory interpretation to understand more about the infinity Laplacian equation, a highly degenerate nonlinear elliptic equation. The PI also intends to characterize asymptotic behaviors of principle eigenfunctions of p-Laplacian operators as p goes to infinity. Other problems concern properties of classical solutions of the infinity Lapalcian equation and uniqueness of absolute minimizers from minimizing more general norms of the gradient. (2) The aim of the weak KAM theory is to use pde approaches to study the Aubry-Mather theory. Our major goal here is to find a variational method to identify the Aubry set. (3) It was known that generalized solutions of Monge-Ampere equations from the optimal mass transfer problems might have singularities. The PI plans to use some tools developed with P. Cannarsa to explore the regularity of the set of singularities. (1) Equations involving the infinity Laplacian operator are very different from elliptic PDEs that people knew before. On one hand, they are second order. On the other hand, the infinity Laplacian operator is so degenerate that those equations sometimes behave as first order PDEs, for example, their solutions even possess some sort of characteristics. Proposed problems in this topic require new methods and ideas which will enhance people's knowledge of elliptic PDEs. Beside its extreme mathematical interest, the infinity Laplacian operator also has important applications in practical issues, for example, to restore images with poor dynamical range, to determine the optimal strategy in the tug-of-war game which is applicable to economic and political modeling, etc. (2) Very little has been known about the structure of the Aubry-Mather set when the dimension is bigger than two. The research proposed in the weak KAM theory part may provide a numerical method to approximate the Aubry set. (3) Monge-Ampere equations from optimal transfer problems have interesting applications in meteorology. The semigeostrophic equations from meteorology can be formulated as a coupled Monge-Ampere/transport problem. The results about the set of singularities of generalized solutions of the Monge-Ampere equation should help people understand how fronts arise in large scale weather pattern.

PI提议继续研究与无穷拉普拉斯算子、弱KAM理论和Monge-Ampere方程解的奇异性有关的问题。(1)无穷拉普拉斯算子产生于最小化梯度的L-无穷范数和两个人的微分游戏称为？拔河？PI打算解决几个问题，从？拔河比赛游戏.其中一个重要的问题是，看看我们如何能够使用博弈论的解释，以了解更多关于无穷大拉普拉斯方程，高度退化的非线性椭圆方程。PI还试图刻画p-Laplacian算子的主本征函数在p趋于无穷大时的渐近行为。 其他问题涉及的性质的经典解决方案的无穷Lapalcian方程和唯一性的绝对极小最小化更普遍的规范的梯度。(2)弱KAM理论的目的是用PDE方法研究Aubry-Mather理论。我们的主要目标是找到一个变分方法来确定奥布里集。(3)已知最优传质问题的Monge-Ampere方程的广义解可能具有奇异性。PI计划使用与P. Cannarsa一起开发的一些工具来探索奇点集的规律性。(1)涉及无穷拉普拉斯算子的方程与人们以前知道的椭圆偏微分方程有很大的不同。一方面，他们是第二秩序。另一方面，无穷Laplacian算子是如此的退化，以至于这些方程有时表现为一阶偏微分方程，例如，它们的解甚至具有某种特征。 提出的问题，在这个主题中需要新的方法和思想，这将提高人们的椭圆偏微分方程的知识。无穷Laplacian算子除了具有极高的数学意义外，在实际问题中也有重要的应用，例如，用于恢复动态范围较差的图像，确定适用于经济和政治建模的拔河游戏中的最优策略等。（2）当维数大于2时，Aubry-Mather集的结构知之甚少。弱KAM理论部分的研究为Aubry集的近似提供了一种数值方法。(3)最优传输问题的Monge-Ampere方程在气象学中有有趣的应用。气象学中的半地转方程可以表述为耦合的Monge-Ampere/输运问题。关于Monge-Ampere方程广义解的奇点集的结果有助于人们理解大尺度天气型中锋面的产生。