Loewner and Loewner-Kufarev equations

Prerequisites: MAT231, together with the courses MAT211, MAT214, MAT234, which can be taken in parallel with the project.

Abstract: The project is aimed at the study and deduction of the Loewner-Kufarev equations in partial derivatives and of its characteristic equation in ordinary derivatives.

Description: The project includes topics from Complex Analysis and Differential Equations. One of the famous descriptions of domain growth is the so-called Loewner subordination chain and its analytic form given by the Loewner equation. Students will be asked to follow the deduction of this equation in partial derivatives. One of the problems to study is the solution of this 1st order PDE by means of the characteristic method. The first part is simple: the solution (given on non-characteristic initial conditions) always exists. The non-trivial part is that it is not always univalent. Students will study the cases when the solution exists and is always univalent.

References:

[1] Ch.Pommerenke, Univalent functions, Vandenhoeck et. Ruprecht GmbH et. Co KG, 1974. (MR)
[2] P.Duren, Univalent Functions, Springer-Verlag, New York, 1983.