Mathematical Sciences: Isoperimetric and Symmetrization Problems

数学科学：等周和对称化问题

• 批准号: 9414149
• 负责人: Richard Laugesen
• 金额: $ 3.83万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-06-01 至 1997-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9414149&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Isoperimetric; Symmetrization; Problems

9414149 Laugesen This work concerns methods for estimating physical constants related to domains in space. For most domains, the constants, such as electrostatic capacity, are impossible to compute. However, symmetry methods such as those employed in this research seek to determine the shape of domains where the extremal values must occur. The problem then shifts to one of computing the constant for the extremal domain - which often can be done explicitly. Work on this project focuses on three groups of problems. The first is to show that for a set in n-dimensional space with a fixed moment of inertia, the Newtonian capacity is minimal when the set is a ball. The analogous results for logarithmic capacity and for the moment of inertia in dimension equal to two has been established. The second isoperimetric problem considered is that of minimizing the ratio of logarithmic to hyperbolic capacity for a continuum in a disc which lies in a fixed concentric subdisc. A solution to this problem would give an upper estimate for hyperbolic capacity in terms of easier-to-compute logarithmic capacity. The third group of problems concerns harmonic mappings of a disc to itself. It is believed that, when suitably normalized, the coefficients of power series expansions of components of such mappings have sharp bounds. Though the first unnormalized coefficient is believed to be less than 5/2, the only estimate know so far is greater than 55. Classical function theory has introduced geometric techniques which have proved to have far wider applicability than first imagined. In this project, the methods of symmetrization, variational techniques and geometric reasoning lead to results of valuable physical significance. ***

9414149劳格森这项工作涉及估计与空间区域有关的物理常数的方法。对于大多数域，诸如静电容量之类的常量是不可能计算的。然而，对称方法，如那些在本研究中使用的方法，试图确定极值必须出现的区域的形状。然后，问题转移到计算极值区域的常量--这通常可以显式完成。这个项目的工作重点是三组问题。首先证明了对于n维空间中具有固定转动惯量的集合，当该集合是球时，牛顿容量是最小的。建立了对数容量和转动惯量等于2的相似结果。所考虑的第二个等周问题是使位于固定同心子圆盘中的圆盘中的连续统的对数容量与双曲线容量之比最小的问题。这个问题的解决方案将给出更容易计算的对数容量方面的双曲容量的上限估计。第三组问题涉及圆盘到其自身的调和映射。人们认为，当适当地归一化时，这类映射的分量的幂级数展开系数有明确的界。尽管第一个未归一化系数据信小于5/2，但到目前为止所知的唯一估计是大于55。经典函数理论引入了几何技术，事实证明，这些技术的适用性比最初想象的要广泛得多。在这个项目中，对称化的方法、变分技术和几何推理方法导致了有价值的物理意义的结果。***