Mathematical Sciences: Isoperimetric and Symmetrization Problems

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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. ***