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

L-无限变分问题是一个寻求泛函的最大值(或最小值)的问题，泛函是一个涉及函数及其梯度的逐点行为的表达式。对这类问题的研究最近变得非常活跃，该项目将支持对该领域一些重要的未决问题的研究。一个特别有趣的问题是变分问题的极小值与相应的Aronsson方程的解之间的关系。其他问题包括Aronsson方程解的唯一性和正则性以及无穷拉普拉斯算子的主本征值的刻画。这些变分问题不仅在数学上很有趣，而且出现在许多不同的应用领域。这些包括在化疗、图像分析和重建中确定最佳放射治疗，以及在某些类型的游戏中确定制胜策略。根据这项研究获得的结果将有助于描述这些应用的数学模型。这是与德克萨斯大学奥斯汀分校的于一峰博士合作的奖项。

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 Yifeng Yu of the University of Texas at Austin.