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

Yuthe Pi建议继续研究与无穷拉普拉斯算子、弱Kam理论和Monge-Ampere方程解的奇性有关的问题。(1)无穷拉普拉斯算子是由最小化梯度的L无穷范数和一个称为拔河的二人微分对策产生的。PI打算解决拔河比赛中的几个问题。游戏。其中一个重要的问题是看我们如何用博弈论的解释来更多地了解无限拉普拉斯方程，一个高度退化的非线性椭圆型方程。PI还刻画了p-拉普拉斯算子的主特征函数在p趋于无穷大时的渐近行为。其他问题涉及无穷拉帕尔西亚方程经典解的性质，以及从最小化更一般的梯度范数到绝对极小值的唯一性。(2)弱KAM理论的目的是用PDE方法研究Aubry-Mather理论。我们这里的主要目标是找到一种识别Aubry集的变分方法。(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共同开发的一些工具来探索奇点集合的规律性。(1)涉及无穷拉普拉斯算子的方程与人们以前所知道的椭圆型偏微分方程组有很大的不同。一方面，他们是二流的。另一方面，无穷拉普拉斯算子是如此退化，以至于这些方程有时表现为一阶偏微分方程组，例如，它们的解甚至具有某种特征。本课题提出的问题需要新的方法和思路来提高人们对椭圆型偏微分方程组的认识。除了极具数学意义外，无穷拉普拉斯算子在实际问题中也有重要的应用，如恢复动态范围较小的图像，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Aubry-Mather集的维度大于2时，人们对其结构知之甚少。弱KAM理论部分的研究可以提供一种近似Aubry集的数值方法。(3)由最优传输问题得到的Monge-Ampere方程在气象学中有着重要的应用。气象学中的半地转方程可以表示为一个耦合的Monge-Ampere/输运问题。关于Monge-Ampere方程广义解的奇性集的结果应该有助于人们理解大尺度天气形势下锋面的产生。

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.