Boundary properties of planar harmonic mappings.
Paul Greiner · Deep Blue (University of Michigan) · 1996
A planar harmonic mapping of a domain ${\rm I\!D}\subset\rm\doubc$ is a complex-valued univalent function which satisfies Laplace's equation $\Delta f = 0$. These include the conformal mappings but their real and imaginary parts do not necessarily satisfy the Cauchy-Riemann equations. This thesis examines three main problems concerning harmonic mappings: determining the geometry of mappings produced with the "shear construction" of Clunie and Sheil-Small, finding the relationship between the boundary function and the dilatation $\omega = \bar f\sb{\bar z}/f\sb{z}$, of the harmonic extension f, and solving linear extremal problems over the family of harmonic self-mappings of the disk. For the mappings produced by the shear construction, relationships between the geometry of the mapping and its dilatation are suggested by calculating examples and displaying them graphically with Mathematica. An integral representation for the shear operator shows that the boundary functions of harmonic mappings with $\vert\omega(z)\vert$ = 1 for z on the boundary are piecewise either constant or concave, depending on the preliminary conformal mapping and the direction of the shear. Since only local properties are involved, this result for harmonic mappings convex in one direction extends to more general harmonic functions. Mappings onto bounded convex domains can be represented by the Poisson integral of the boundary function. As with the shear construction, choosing simple examples suggests relationships between the boundary function and the dilatation. An examination of the Poisson integral shows that if the boundary function has a jump-discontinuity at a point z, then $\vert\omega(z)\vert$ = 1 and the value of $\omega(z)$ depends only on the right- and left-hand limits of the boundary function at z. With some additional smoothness, a similar calculation shows that if the derivative of the boundary function has a jump-discontinuity at a point z, then $\vert\omega(z)\vert$ = 1 and the value of $\omega(z)$ depends only on the value of the boundary function at z. The Poisson integral representation reduces linear extremal problems for self-mappings of the disk to those of maximizing a functional over admissible boundary correspondences. The problems of maximizing a balanced sum of Fourier coefficients and of maximizing $\vert\alpha f(z\sb1) + f(z\sb2)\vert$ for $z\sb1$ and $z\sb2$ fixed points in the unit disk are discussed in detail.