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

l -无穷变分问题是一个人试图找到一个泛函的最大值（或最小值）的问题，这个泛函是一个涉及函数的逐点行为及其梯度的表达式。对这些问题的研究最近变得非常活跃，这个项目将支持研究该领域的一些重要的开放性问题。特别感兴趣的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特征值的表征。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。本研究的结果将有助于描述这些应用的数学模型。这是我与德克萨斯大学奥斯汀分校的余易峰博士共同获得的奖项。

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.