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On the Geometric Properties of the Poisson Kernel for the Lamé Equation

Abstract

It is shown that the Poisson kernel for the Lamé equation in a disk can be interpreted as a bi-univalent mapping of the projection of an elliptic cone onto the projection of the surface of revolution of a hyperbola. The corresponding mapping \({{f}_{\sigma }}\) of these surfaces is bijective. Such an interpretation sheds light on the nature of the well-known special property of solutions of elliptic systems on a plane to map points to curves and vice versa. In particular, a singular point of the kernel under study can be considered as the projection of the cone element so that the mapping \({{f}_{\sigma }}\) is regular in the sense that this element is bijectively mapped into a curve.

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ACKNOWLEDGMENTS

I am grateful to V.I. Vlasov for fruitful discussions and valuable advices when preparing the article.

Funding

This work was supported by the Russian Foundation for Basic Research (project no. 18-01-00764) and the Ministry of Education and Science of the Russian Federation (project no. 1.3843.2017/4.6).

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Correspondence to A. O. Bagapsh.

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Translated by E. Chernokozhin

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Bagapsh, A.O. On the Geometric Properties of the Poisson Kernel for the Lamé Equation. Comput. Math. and Math. Phys. 59, 2124–2144 (2019). https://doi.org/10.1134/S0965542519120042

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  • DOI: https://doi.org/10.1134/S0965542519120042

Keywords:

  • elliptic systems
  • Lamé equation
  • non-univalent mappings