Schwarz–Christoffel Asymptotic Solution of the Loewner Equation

Abstract

The paper is devoted to asymptotic solutions of the chordal Loewner differential equation which map the upper half-plane $\mathbb{H}$ onto $\mathbb{H}\setminus\Gamma$, $\Gamma=\Gamma_{1}\cup\Gamma_{2}$, where $\Gamma_{1}$ is a segment on the imaginary axis and $\Gamma_{2}$ is a horizontal segment through the upper endpoint of $\Gamma_{1}$, symmetric with respect to the imaginary axis. The mapping is represented by a Schwarz–Christoffel integral whose accessory parameters are to be defined. We reduce the problem to solving three equations and give asymptotical expansions, in terms of the Loewner time parameter, for accessory parameters and driving functions in a neighborhood of the critical point $\Gamma_{1}\cap\Gamma_{2}$. The driving function in the Loewner differential equation is a combination of a constant function and a square root function.