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.

莱昂·西蒙（Leon Simon）和布莱恩·怀特（Brian White）将研究与其他几何变异问题的最小表面和极值的规律性和奇异性有关的问题。 西蒙的研究将沿三个广泛的方向进行。其中的第一个涉及与奇异表面相关的问题。第二个涉及函数图最小表面方程的整个和外部解。最后一个主题涉及Willmore型功能。 有关奇异表面的单数问题的基本问题与此类表面在接近奇异点的方法上的行为有关。在孤立的奇异性并现在将研究更普遍的情况下，西蒙已经获得了强劲的结果。对最小表面方程的研究将围绕整个和外部溶液的渐近行为。在这里，重点将是在高维度的调查上。 Willmore功能是平均曲率的一定平均值。 这里的问题与在某个属的表面上最小化了这种功能有关。 怀特将研究最小化积分电流的奇异区域的尺寸。这可能会导致发现此类分数维度。一个相关的问题是找到条件，几乎每个边界都会导致规则面积最小化磁盘。他还将调查riemannian歧管中紧凑的最小亚曼叶的存在。一个研究方向将涉及对存在多少个嵌入式最小亚曼叶的研究。