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


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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  1. L. D. Landau and E. M. Lifshitz, Theory of Elasticity (Butterworth-Heinemann, Oxford, 1986; Nauka, Moscow, 1987).

  2. Hua Loo-Keng, Lin Wei, and Wu Ci-Quian, Second-Order Systems of Partial Differential Equations in the Plane (Pitman, Boston, 1985).

  3. A. O. Bagapsh and K. Yu. Fedorovskiy, “C1-approximation of functions by solutions of second-order elliptic systems on compact sets in \({{\mathbb{R}}^{2}}\),” Proc. Steklov Inst. Math. 298, 35–50 (2017).

    Article  Google Scholar 

  4. I. G. Petrovskii, “On analyticity of solutions to systems of partial differential equations,” Mat. Sb. 5, 3–70 (1939).

    Google Scholar 

  5. M. I. Vishik, “On strongly elliptic systems of differential equations,” Mat. Sb. 29, 615–676 (1951).

    MathSciNet  Google Scholar 

  6. A. O. Bagapsh, “Green’s function and Poisson integral in a circle disk for strongly elliptic systems with constant coefficients,” Vestn. Mosk. Gos. Tekh. Univ. im. N.E. Baumana, Ser. Estestv. Nauki, No. 6, 4–18 (2017).

    Google Scholar 

  7. P. Duren, Harmonic Mappings in the Plane (Cambridge Univ. Press, Cambridge, 2004).

    Book  Google Scholar 

  8. T. Rado, “Aufgabe 41,” Jahresber. Deutsch. Math.-Verein. 35, 49 (1926).

    Google Scholar 

  9. H. Kneser, “Lösung der Aufgabe 41,” Jahresber. Deutsch. Math.-Verein. 35, 123–124 (1926).

    Google Scholar 

  10. J. G. Clunie and T. Sheil-Small, “Harmonic univalent functions,” Ann. Acad. Sci. Fenn. Ser. A I Math. 9, 3–25 (1984).

  11. W. Hengartner and G. Schober, “Univalent harmonic functions,” Trans. Am. Math. Soc. 299, 1–31 (1987).

    MathSciNet  Article  Google Scholar 

  12. D. Bshouty, W. Hengartner, and O. Hossian, “Harmonic typically real mappings,” Math. Proc. Cambridge Philos. Soc. 119, 673–680 (1996).

    MathSciNet  Article  Google Scholar 

  13. P. Duren, W. Hengartner, and R. S. Laugesen, “The argument principle for harmonic functions,” Am. Math. Mon. 103, 411–415 (1996).

    MathSciNet  Article  Google Scholar 

  14. S. I. Bezrodnykh and V. I. Vlasov, “On a problem in the constructive theory of harmonic mappings,” J. Math. Sci. 201 (6), 705–732 (2014).

    MathSciNet  Article  Google Scholar 

  15. D. Bshouty, S. S. Joshi, and S. B. Joshi, “On close-to-convex harmonic mappings,” Complex Var. Elliptic Equations 58 (9), 1195–1199 (2013).

    MathSciNet  Article  Google Scholar 

  16. D. Kalaj, S. Ponnusamy, and M. Vuorinen, “Radius of close-to-convexity and fully starlikeness of harmonic mappings,” Complex Var. Elliptic Equations 59 (4), 539–552 (2014).

    MathSciNet  Article  Google Scholar 

  17. G. Alessandrini and V. Nesi, “Elliptic systems and material interpenetration,” Funct. Approx. Comment. Math. 40 (1), 105–115 (2009).

    MathSciNet  Article  Google Scholar 

  18. A. B. Zaitsev, “Mappings by the solutions of second-order elliptic equations,” Math. Notes 95 (5–6), 642–655 (2014).

    MathSciNet  Article  Google Scholar 

  19. A. B. Zaitsev, “On univalence of solutions of second-order elliptic equations in the unit disk on the plane,” J. Math. Sci. 215 (5), 601–607 (2015).

    MATH  Google Scholar 

  20. A. A. Savelov, Plane Curves: Systematics, Properties, and Applications (Fizmatlit, Moscow, 1960) [in Russian].

    Google Scholar 

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


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).

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  • elliptic systems
  • Lamé equation
  • non-univalent mappings