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On Dirichlet to Neumann and Robin to Neumann operators suitable for reflecting harmonic functions subject to a non-homogeneous condition on an arc

Abstract

According to the Schwarz symmetry principle, every harmonic function vanishing on a real-analytic curve has an odd continuation, while a harmonic function satisfying homogeneous Neumann condition has an even continuation. Using a technique of Dirichlet to Neumann and Robin to Neumann operators, we derive reflection formulae for non-homogeneous Neumann and Robin conditions from a reflection formula subject to a non-homogeneous Dirichlet condition.

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Acknowledgements

The authors are very grateful to the anonymous referee for the suggestions, which helped to improve the paper. The second author would like to thank the organizers of the International Conference on Complex Analysis, Potential Theory and Applications in honour of Professor Stephen J Gardiner on the occasion of his 60th Birthday and Professor Lucian Beznea for providing the authors with his interesting papers.

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Correspondence to Tatiana Savina.

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Aldawsari, M., Savina, T. On Dirichlet to Neumann and Robin to Neumann operators suitable for reflecting harmonic functions subject to a non-homogeneous condition on an arc. Anal.Math.Phys. 9, 729–745 (2019). https://doi.org/10.1007/s13324-019-00314-w

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  • DOI: https://doi.org/10.1007/s13324-019-00314-w

Keywords

  • Schwarz symmetry principle
  • Dirichlet to Neumann operator
  • Robin to Neumann operator
  • Analytic continuation