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On the Maximum Principle for Solutions of Second Order Elliptic Equations

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Abstract

In this paper, we found sufficient conditions, under fulfillment of which the maximum principle for the solution of a second order partial differential elliptic equation in the unit circle meets maximum principle. It is proved that if a quasiconformality coefficient of such function satisfies certain boundary conditions, then this function meets maximum principle. While proving the main result, we use integral representations of solutions of this equation and properties of the Cauchy type integral and functions of Hardy and Smirnoff classes.

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REFERENCES

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  2. Zaitsev, A.B. “On the univalence of solutions of second order elliptic equations in the unit disk on the plane”, J. Math. Sci. (N. Y.) 215 (5), 601–607 (2016).

  3. Bagapsh, A.O. “The Poisson integral and Green's function for one strongly elliptic system of equations in a circle and an ellipse”, Comput. Math. Math. Phys. 56 (12), 2035–2042 (2016).

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  5. Danilyuk, I.I. Non-regular boundary value problems on the plane (Nauka, Moscow, 1975) [in Russian].

  6. Zaitsev, A.B. “Mappings by the Solutions of Second-Order Elliptic Equations”, Math. Notes 95 (5), 642–655 (2014).

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ACKNOWLEDGMENTS

The author would like to express his sincere gratitude to P.V. Paramonov and K.Yu. Fedorovsky for their attention to the paper and valuable remarks.

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Authors and Affiliations

  1. Russian Technological University MIREA, 78 Vernadsky Ave., 119454, Moscow, Russia

    A. B. Zaitsev

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  1. A. B. Zaitsev
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Correspondence to A. B. Zaitsev.

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Russian Text © The Author(s), 2020, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2020, No. 8, pp. 11–17.

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Zaitsev, A.B. On the Maximum Principle for Solutions of Second Order Elliptic Equations. Russ Math. 64, 8–13 (2020). https://doi.org/10.3103/S1066369X20080022

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  • DOI: https://doi.org/10.3103/S1066369X20080022

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