Abstract
Let I be a line segment in the complex plane \(\mathbb C\). We describe a method of constructing a bi-Lipschitz sense-preserving mapping of \(\mathbb C\) onto itself, which is harmonic in \(\mathbb C\setminus I\) and coincides with a given sufficiently regular function \(f:I\rightarrow \mathbb C\). As a result we show that a quasiconformal self-mapping of \(\mathbb C\) which is harmonic in \(\mathbb C\setminus I\) does not have to be harmonic in \(\mathbb C\).
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Grigoryan, A., Michalski, A. & Partyka, D. Extensions of Harmonic Functions of the Complex Plane Slit Along a Line Segment. Potential Anal 61, 65–81 (2024). https://doi.org/10.1007/s11118-023-10103-7
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DOI: https://doi.org/10.1007/s11118-023-10103-7