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Representation of the Green’s Function of the Dirichlet Problem for the Polyharmonic Equation in the Ball

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Abstract

We define the elementary solution of the polyharmonic equation, with the help of which an explicit representation of the Green’s function of the Dirichlet problem for the polyharmonic equation in the unit ball is given for all space dimensions except for some finite set. On the basis of the obtained Green’s function, the solution of the homogeneous Dirichlet problem in the unit ball is constructed. As an example, an explicit form of the solution of the homogeneous Dirichlet problem for the inhomogeneous polyharmonic equation with the simplest polynomial right-hand side is found.

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REFERENCES

  1. Begehr, H., Biharmonic Green functions, Le Matematiche, 2006, vol. 61, pp. 395–405.

    MathSciNet  MATH  Google Scholar 

  2. Begehr, H. and Vaitekhovich, T., Modified harmonic Robin function, Complex Var. Elliptic Equat., 2013, vol. 58, no. 4, pp. 483–496.

    Article  MathSciNet  MATH  Google Scholar 

  3. Sadybekov, M.A., Torebek, B.T., and Turmetov, B.Kh., On an explicit form of the Green function of the Robin problem for the Laplace operator in a circle, Adv. Pure Appl. Math., 2015, vol. 6, no. 3, pp. 163–172.

    Article  MathSciNet  MATH  Google Scholar 

  4. Karachik, V.V. and Turmetov, B.Kh., On Green’s function of the Robin problem for the Poisson equation, Adv. Pure Appl. Math., 2019, vol. 10, no. 3, pp. 203–214.

    Article  MathSciNet  MATH  Google Scholar 

  5. Ying Wang and Liuqing Ye, Biharmonic Green function and biharmonic Neumann function in a sector, Complex Var. Elliptic Equat., 2013, vol. 58, no. 1, pp. 7–22.

    Article  MathSciNet  MATH  Google Scholar 

  6. Ying Wang, Tri-harmonic boundary value problems in a sector, Complex Var. Elliptic Equat., 2014, vol. 59, no. 5, pp. 732–749.

    Article  MathSciNet  MATH  Google Scholar 

  7. Boggio, T., Sulle funzioni di Green d’ordine \(m \), Palermo Rend., 1905, vol. 20, pp. 97–135.

    Article  MATH  Google Scholar 

  8. Kalmenov, T.Sh., Koshanov, B.D., and Nemchenko, M.Y., Green function representation for the Dirichlet problem of the polyharmonic equation in a sphere, Complex Var. Elliptic Equat., 2008, vol. 53, pp. 177–183.

    Article  MathSciNet  MATH  Google Scholar 

  9. Kal’menov, T.S. and Suragan, D., On a new method for constructing the Green function of the Dirichlet problem for the polyharmonic equation, Differ. Equations, 2012, vol. 48, no. 3, pp. 441–445.

    Article  MathSciNet  MATH  Google Scholar 

  10. Karachik, V.V., Dirichlet and Neumann boundary value problems for the polyharmonic equation in the unit ball, Mathematics, 2021, vol. 9, no. 16, p. 1907.

    Article  Google Scholar 

  11. Karachik, V.V., Green’s function of Dirichlet problem for biharmonic equation in the ball, Complex Var. Elliptic Equat., 2019, vol. 64, no. 9, pp. 1500–1521.

    Article  MathSciNet  MATH  Google Scholar 

  12. Karachik, V.V., The Green function of the Dirichlet problem for the biharmonic equation in a ball, Comput. Math. Math. Phys., 2019, vol. 59, no. 1, pp. 66–81.

    Article  MathSciNet  MATH  Google Scholar 

  13. Karachik, V.V., The Green function of the Dirichlet problem for the triharmonic equation in the ball, Math. Notes, 2020, vol. 107, no. 1, pp. 105–120.

    Article  MathSciNet  MATH  Google Scholar 

  14. Karachik, V.V. and Torebek, B.T., On the Dirichlet–Riquier problem for biharmonic equations, Math. Notes, 2017. T. 102, no. 1, pp. 31–42.

  15. Karachik, V.V., A Neumann-type problem for the biharmonic equation, Sib. Adv. Math., 2017, vol. 27, no. 2, pp. 103–118.

    Article  MathSciNet  Google Scholar 

  16. Soldatov, A.P., On the Fredholm property and index of the generalized Neumann problem, Differ. Equations, 2020, vol. 56, no. 2, pp. 212–220.

    Article  MathSciNet  MATH  Google Scholar 

  17. Karachik, V.V., Green’s functions of the Navier and Riquier–Neumann problems for the biharmonic equation in the ball, Differ. Equations, 2021, vol. 57, no. 5, pp. 654–668.

    Article  MathSciNet  MATH  Google Scholar 

  18. Sweers, G., A survey on boundary conditions for the biharmonic, Complex Var. Elliptic Equat., 2009, vol. 54, pp. 79–93.

    Article  MathSciNet  MATH  Google Scholar 

  19. Karachik, V.V., Riquier–Neumann problem for the polyharmonic equation in a ball, Differ. Equations, 2018. v. 54, no. 5, pp. 648–657.

  20. Karachik, V.V., The Green function of the Navier problem for the polyharmonic equation in a ball, J. Math. Sci., 2023, vol. 269, no. 2, pp. 189–204.

    Article  MathSciNet  Google Scholar 

  21. Karachik, V.V., Riquier–Neumann problem for the polyharmonic equation in a ball, Mathematics, 2023, vol. 11, no. 4, p. 1000.

    Article  Google Scholar 

  22. Karachik, V., Turmetov, B., and Yuan, H., Four boundary value problems for a nonlocal biharmonic equation in the unit ball, Mathematics, 2022, vol. 10, no. 7, p. 1158.

    Article  Google Scholar 

  23. Begehr, H., Burgumbayeva, S., and Shupeyeva, B., Remark on Robin problem for Poisson equation, Complex Var. Elliptic Equat., 2017, vol. 62, no. 10, pp. 1589–1599.

    Article  MathSciNet  MATH  Google Scholar 

  24. Akel, M. and Begehr, H., Neumann function for a hyperbolic strip and a class of related plane domains, Math. Nachrichten, 2017, vol. 290, no. 4, pp. 490–506.

    Article  MathSciNet  MATH  Google Scholar 

  25. Lin, H., Harmonic Green and Neumann functions for domains bounded by two intersecting circular arcs, Complex Var. Elliptic Equat., 2020, vol. 67, pp. 79–95.

    Article  MathSciNet  MATH  Google Scholar 

  26. Begehr, H., Burgumbayeva, S., Dauletkulova, A., and Lin, H., Harmonic Green functions for the Almaty apple, Complex Var. Elliptic Equat., 2020, vol. 65, no. 11, pp. 1814–1825.

    Article  MathSciNet  MATH  Google Scholar 

  27. Grebenkov, D.S. and Traytak, S.D., Semi-analytical computation of Laplacian Green functions in three-dimensional domains with disconnected spherical boundaries, J. Comput. Phys., 2019, vol. 379, pp. 91–117.

  28. Hsu, C.-W. and Hwu, C., Green’s functions for unsymmetric composite laminates with inclusions, Proc. R. Soc. A: Math. Phys. Eng. Sci., 2020, vol. 476, no. 2233, p. 20190437.

    Article  MathSciNet  MATH  Google Scholar 

  29. Bitsadze, A.V., Uravneniya matematicheskoi fiziki (Equations of Mathematical Physics), Moscow: Nauka, 1982.

  30. Begerh, H., Vu, T.N.H., and Zhang, Z.-X., Polyharmonic Dirichlet problems, Proc. Steklov Inst. Math., 2006, vol. 255, pp. 13–34.

    Article  MathSciNet  Google Scholar 

  31. Sobolev, S.L., Vvedenie v teoriyu kubaturnykh formul (Introduction to the Theory of Cubature Formulas), Moscow: Nauka, 1974.

    Google Scholar 

  32. Vladimirov, V.S., Uravneniya matematicheskoi fiziki (Equations of Mathematical Physics), Moscow: Nauka, 1981.

    Google Scholar 

  33. Gazzola, F., Grunau, H.C., and Sweers, G., Polyharmonic Boundary Value Problems, Berlin: Springer, 2010.

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Authors and Affiliations

  1. South Ural State University, Chelyabinsk, 454080, Russia

    V. V. Karachik

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  1. V. V. Karachik
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Correspondence to V. V. Karachik.

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Translated by V. Potapchouck

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Karachik, V.V. Representation of the Green’s Function of the Dirichlet Problem for the Polyharmonic Equation in the Ball. Diff Equat 59, 1061–1074 (2023). https://doi.org/10.1134/S0012266123080050

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  • DOI: https://doi.org/10.1134/S0012266123080050

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