Bounded connected components of polynomial lemniscates, polynomial approximations of Z-bar, and applications to torsional rigidity.

The work in this dissertation lies at the intersection of functional analysis, numerical analysis, and approximation theory. A well studied topic in mechanics and classical elasticity theory is that of torsional rigidity. Extremal properties are often of interest in practical problems, thus we consider how the geometric structure of a region influences its torsional rigidity. Much of the work here is focused on finding optimal or near optimal solutions to isoperimetric-type problems. Motivation for finding these solutions stems from Pólya’s conjecture for the torsional rigidity of polygons. Additional applications related to principal frequency and logarithmic capacity are mentioned. One may express the torsional rigidity of a simply connected and bounded Jordan domain by a variational form of the Hadamard type. In this manner, we discuss how symmetrization techniques may be exploited to provide further understanding of the torsional rigidity of various regions. Finally we provide upper bound numerics in support of Pólya’s conjecture for the torsional rigidity of pentagons. We calculate high accuracy numerical approximations among all equilateral pentagons with fixed area which verifies that the maximizer of torsional rigidity is within less than one third of one degree of the regular pentagon. Additionally, we investigate the Bergman analytic content of various simply connected and bounded Jordan domains. The square of the Bergman analytic content of a simply connected region is equal to that region’s torsional rigidity. We give examples where this may be calculated directly and classify regions where the Bergman projection of ¯z is a polynomial of a certain form. By considering families of polynomial lemniscates in the complex plane that include a bounded connected component we may calculate the Bergman projection of ¯z exactly. We investigate the necessary and sufficient conditions required for various polynomial lemniscates to have such a property. A region whose boundary is the level set of such a polynomial lemniscate may then have its torsional rigidity calculated directly through the Bergman projection of ¯z. In conclusion, various open problems are discussed along with several possible directions for future work.