Loewner equation for Laplacian growth: A Schwarz-Christoffel-transformation approach

Abstract

The problem of Laplacian growth is considered within the Loewner-equation framework. A new method of deriving the Loewner equation for a large class of growth problems in the half-plane is presented. The method is based on the Schwarz-Christoffel transformation between the so-called `mathematical planes' at two infinitesimally separated times. Our method not only reproduces the correct Loewner evolution for the case of slit-like fingers but also can be extended to treat more general growth problems. In particular, the Loewner equation for the case of a bubble growing into the half-plane is presented.