Extensions of Harmonic Functions of the Complex Plane Slit Along a Line Segment

Abstract Let I be a line segment in the complex plane $$\mathbb C$$ C . We describe a method of constructing a bi-Lipschitz sense-preserving mapping of $$\mathbb C$$ C onto itself, which is harmonic in $$\mathbb C\setminus I$$ C \ I and coincides with a given sufficiently regular function $$f:I\rightarrow \mathbb C$$ f : I → C . As a result we show that a quasiconformal self-mapping of $$\mathbb C$$ C which is harmonic in $$\mathbb C\setminus I$$ C \ I does not have to be harmonic in $$\mathbb C$$ C .