Mathematical Sciences: Geometric Variational Problems and Related PDE Questions

数学科学：几何变分问题及相关偏微分方程问题

• 批准号: 8703537
• 负责人: Leon Simon
• 金额: $ 29.45万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1987
• 资助国家: 美国
• 起止时间: 1987-07-01 至 1990-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8703537&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Geometric; Variational; Problems

Leon Simon and Brian White will study problems relating to regularity and singularity of minimal surfaces and extrema of other geometric variational problems. Simon's research will proceed in three broad directions. The first of these involves problems relating to the singular set of minimal surfaces. The second concerns entire and exterior solutions of the minimal surface equation for the graph of a function. The final topic involves Willmore-type functionals. The basic questions regarding singular sets of minimal surfaces are concerned with the behaviour of such surfaces on approach to a singular point. Simon has already obtained strong results in the case where the singularity is isolated and will now investigate more general situations. The investigations of the minimal surface equation will center around the asymptotic behaviour of entire and exterior solutions. Here the emphasis will be on investigations in a high dimensional setting. The Willmore functional is a certain average of the mean curvature. The questions here relate to minimizing this functional over surfaces of a certain genus. White will investigate the dimension of the singular set of area minimizing integral currents. This may lead to the discovery of such sets of fractional dimension. A related problem is to find conditions under which almost every boundary gives rise to regular area minimizing disks. He will also investigate the existence of compact minimal submanifolds in Riemannian manifolds. One direction of research will involve an investigation of how many embedded minimal submanifolds exist.

里昂·西蒙和布莱恩·怀特将研究与极小曲面的正则性和奇异性以及其他几何变分问题的极值有关的问题。西蒙的研究将从三个大的方向进行。第一个问题涉及与极小曲面的奇异集有关的问题。第二个问题是关于函数图形的极小曲面方程的整体解和外解。最后一个话题涉及威尔莫尔型泛函。关于极小曲面奇异集的基本问题是关于这类曲面在逼近奇点时的行为。Simon已经在奇点孤立的情况下得到了很好的结果，现在将研究更一般的情况。极小曲面方程的研究将围绕整体解和外部解的渐近行为展开。在这里，重点将放在高维度环境中的调查上。威尔莫尔泛函是平均曲率的某个平均值。这里的问题涉及在某一亏格的曲面上最小化这个泛函。怀特将研究使积分电流最小化的面积奇异集的维度。这可能导致发现这样的分数维集。一个相关的问题是找到几乎每个边界都产生规则面积最小化圆盘的条件。他还将研究黎曼流形中紧致极小子流形的存在性。一个研究方向将涉及调查存在多少嵌入的极小子流形。