Mathematical Sciences: Classical Complex Analysis

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ABSTRACT Proposal: DMS-9532087 PI: Marshall Under this grant, Marshall will investigate problems in three areas of analytic function theory. A new approach to constructing good metrics for lower bounds for extremal distance will be applied to the Angular Derivative Problem of Ahlfors 1930 . Secondly, he will investigate the accuracy of a promising new technique for the numerical computation of conformal maps. Thirdly, he will attempt to improve constructions of interpolating functions for the Dirichlet space and its multipliers, using more natural linear combinations of reproducing kernels. Conformal mapping has been used as a tool in science and engineering for many years. One way conformal maps are used is to transform a problem on a complicated region in the complex plane to a related problem on a "standard" region, such as a disk or half-plane, where known techniques can be used. The solution on the standard region is then transformed by the inverse of the conformal map to a solution of the original problem on the original region. Classically, this method was used for problems related to Laplace's equation. For example, temperature at equilibrium on a thin metallic plate satisfies Laplace's equation. There have been numerous non-classical applications of conformal maps developed in the last twenty five years, many of which are not governed by Laplace's equation. There are applications in electro-magnetics, vibrating membranes and acoustics, transverse vibrations and buckling of plates, elasticity, heat transfer, and fluid flow for example. While early applications used explicit analytic representations for conformal maps, modern uses require conformal maps of more complicated regions which cannot be represented easily in terms of elementary functions. One must therefore resort to numerical approximations to the desired conformal maps. Marshall will investigate the accuracy of a new technique which rapidly computes conformal maps and their inverses. This technique is fast enough that it can be used for experimentation on a typical workstation. The angular derivative problem he will investigate is about geometrical conditions on the boundary of a region that guarantee that certain conformal maps (defined inside the region) extend to be conformal at a point on the boundary. Dirichlet functions on the disk are not conformal, but the image of the disk has finite area (counting overlaps). They arise from potentials with bounded total energy. The interpolation problems Marshall will study are closely related to constructing a temperature distribution on a thin metal plate with preassigned values at certain points on the plate, and with smallest possible total energy.