Mathematical Sciences: Isoperimetric and Symmetrization Problems

[9414149]劳格森这项工作涉及估计空间中与域有关的物理常数的方法。对于大多数领域，常数，如静电容量，是不可能计算的。然而，对称性方法（如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。然后问题就变成了计算极值域的常数——这通常可以明确地完成。这个项目的工作集中在三组问题上。第一个是证明对于n维空间中具有固定转动惯量的集合，当该集合是一个球时，牛顿容量最小。建立了对数容量和二维惯性矩的类似结果。所考虑的第二个等周问题是在一个固定的同心子圆盘上的连续体的对数容量与双曲容量之比的最小化问题。这个问题的解决方案将根据更容易计算的对数容量给出双曲容量的上限估计。第三组问题涉及圆盘到自身的调和映射。我们认为，当适当地归一化时，这种映射的分量的幂级数展开式的系数具有明显的界。虽然第一个非归一化系数被认为小于5/2，但迄今为止已知的唯一估计值大于55。经典函数理论引入了几何技术，这些技术已被证明具有比最初想象的更广泛的适用性。在这个项目中，对称、变分技术和几何推理的方法导致了有价值的物理意义的结果。＊＊＊

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. ***