One-parametric families of conformal mappings of unbounded doubly connected polygonal domains

Abstract

We propose an approximate method for finding a conformal mapping of a concentric annulus onto an arbitrary unbounded doubly connected polygonal domain. This method is based on ideas related to the parametric Löwner — Komatu method. We consider smooth one-parametric families of conformal mappings $\mathcal{F}(z,t)$ of concentric annuli onto doubly connected polygonal domains $\mathcal{D}(t)$, which are obtained from a fixed unbounded doubly connected polygonal domain $\mathcal{D}$ by making a finite number of rectilinear or, in general, polygonal slits of variable length; at the same time, we do not suppose the monotonicity of family of domains $\mathcal{D}(t)$. The integral representation for the conformal mappings $\mathcal{F}(z,t)$ includes unknown (accessory) parameters. We find a partial differential equation for these families of conformal mappings and derive from it a system of differential equations describing the dynamics of the accessory parameters as the parameter $t$ varies and the dynamics of the conformal modulus of a given doubly connected domain as a function of the parameter $t$. The right hand sides of resulting system of ordinary differential equations include functions being the velocities of the end points of slits. This allows us to control completely the dynamics of slits, in particular, to achieve their consistent change in the case where more than one slit is made in the domain $\mathcal{D}$. Examples illustrating the efficiency of proposed method are provided. We mention that we have already considered the parametric method proposed in this paper but for the case of bounded doubly connected polygonal domains.