Collaborative Research: L-infinity variational problems and the Aronsson equation

L-无穷变分问题是这样的问题，其中人们试图找到一个泛函的最大值（或最小值），该泛函是一个涉及函数及其梯度的逐点行为的表达式。 最近，对这些问题的研究非常活跃，本项目将支持对该领域一些重要的未决问题的研究。一个特别感兴趣的是变分问题的极小值和相应的Aronsson方程的解决方案之间的关系。其他问题包括唯一性和正则性的解决方案的Aronsson方程和表征的主要特征值的无穷拉普拉斯算子。 这些变分问题不仅是有趣的数学，但出现在一些不同的应用领域。这些包括确定化疗中的最佳放射治疗，图像分析和重建以及确定某些类型游戏中的获胜策略。根据这项研究所获得的结果将有助于描述这些应用的数学模型。 这是一个与肯塔基州大学的王昌友博士合作的奖项。

L-infinity variational problems are problems where one seeks to find the maximum (or minimum) of a functional that is an expression involving the pointwise behavior of a function and its gradient. The study of such problems has become very active recently and this project will support the study of a number of important open questions in the area. A particular interest is the relationship between minimizers of the variational problems and solutions of the corresponding Aronsson equation. Other questions include the uniqueness and regularity of solutions of the Aronsson equations and the characterization of the principal eigenvalue of the infinity-Laplacian operator. These variational problems are not only interesting mathematically but arise in a number of different areas of applications. These include the determination of optimal radiation treatments in chemotherapy, in image analysis and reconstruction and in determining winning strategies in certain types of games. The results obtained under this research will help describe the mathematical models of these applications. This is a collaborative award with Dr Changyou Wang of the University of Kentucky.