Skip to main content
SpringerLink
Log in

Cylindrical Poisson kernel method and its applications

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

Foundations of the modified Poisson kernel method were laid out by Finkelstein and Scheinberg in 1975 in the context of explicit solvability for the adaptive Dirichlet problem in a half plane. Over the past decade, this method has been further developed, and new applications have appeared both in the field of harmonic analysis and operator theory and in practical Schrödinger problems. In this paper, we pay special attention to cylindrical Poisson kernel method’s application for the integral representations of harmonic functions in a cylinder.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Agarwal, R.P., Gala, S., Ragusa, M.A.: A regularity criterion of the 3D MHD equations involving one velocity and one current density component in Lorentz space. Z. Angew. Math. Phys. 71, 11 (2020)

    Article  MathSciNet  Google Scholar 

  2. Azarin, V.S.: Generalization of a theorem of Hayman on subharmonic functions in an \(m\)-dimensional cone. Am. Math. Soc. Transl. 80(2), 119–138 (1969)

    MATH  Google Scholar 

  3. Bahrouni, A., Ounaies, H., Rădulescu, V.D.: Infinitely many solutions for a class of sublinear Schrödinger equations with indefinite potentials. Proc. R. Soc. Edinburgh Sect. A 145(3), 445–465 (2015)

    Article  Google Scholar 

  4. Bahrouni, A., Ounaies, H., Rădulescu, V.D.: Bound state solutions of sublinear Schrödinger equations with lack of compactness. Rev. R. Acad. Cienc. Exactas Fís. 113(2), 1191–1210 (2019)

    MathSciNet  MATH  Google Scholar 

  5. Carleman, T.: Propriétés asymptotiques des functions fondamentales des membranes vibrantes. Proc. Skand. Mat. Kongr. Stockholm 1934, 34–44 (1935)

    MATH  Google Scholar 

  6. Carleman, T.: Über die Approximation analytischer Funktionen durch lineare Aggregate von vorgegebenen Potenzen. Ark. för Mat. Astr. Fys. 17, 1–30 (1923)

    MATH  Google Scholar 

  7. Courant, R., Hilbert, D.: Methods of Mathematical Physics, vol. 1. Interscience Publishers, New York (1953)

    MATH  Google Scholar 

  8. Finkelstein, M., Scheinberg, S.: Kernels for solving problems of Dirichlet type in a half-plane. Adv. Math. 18(1), 108–113 (1975)

    Article  MathSciNet  Google Scholar 

  9. Goodrich, S., Ragusa, M.A., Scapellato, A.: Partial regularity of solutions to \(p(x)\)-Laplacian PDEs with discontinuous coefficients. J. Differ. Equ. 268(9), 5440–5468 (2020)

    Article  MathSciNet  Google Scholar 

  10. Guariglia, E.: Riemann zeta fractional derivative-functional equation and link with primes. Adv. Difference Equ. 261, 15 (2019)

    MathSciNet  MATH  Google Scholar 

  11. Guariglia, E.: Fractional Derivative of the Riemann Zeta Function. Fractional Dynamics, pp. 357–368. De Gruyter Open, Berlin (2015)

    Google Scholar 

  12. Guariglia, E.: Fractional calculus, zeta functions and Shannon entropy. Open Math. 19(1), 87–100 (2021)

    Article  MathSciNet  Google Scholar 

  13. Guariglia, E., Tamilvanan, K.: On the stability of radical septic functional equations. Mathematics 12(8), 2229 (2020)

    Article  Google Scholar 

  14. Guliyev, V.S., Guliyev, R.V., Omarova, M.N., Ragusa, M.A.: Schrödinger type operators on local generalized Morrey spaces related to certain nonnegative potentials. Discrete Contin. Dyn. Syst. Ser. A 25(2), 671–690 (2020)

    MathSciNet  MATH  Google Scholar 

  15. Kassay, G., Rădulescu, V.D.: Equilibrium Problems and Applications. Mathematics in Science and Engineering. Elsevier/Academic Press, London (2018)

    MATH  Google Scholar 

  16. Kheyfits, A.: The Dirichlet problem in a half-space for the Schrödinger operator with boundary data of arbitrary growth at infinity. (Russian). Dokl. Akad. Nauk 325(5), 937–939 (1992)

    Google Scholar 

  17. Kheyfits, A.: Dirichlet problem for the Schrödinger operator in a half-space with boundary data of arbitrary growth at infinity. Differ. Integral Equ. 10(1), 153–164 (1997)

    MathSciNet  MATH  Google Scholar 

  18. Levin, B., Kheyfits, A.: Asymptotic Behavior of Subfunctions of Time-independent Schrödinger Operators. Mathematical Society Monographs Series, vol. 11, pp. 323–397. Science Press, Beijing (2008)

    MATH  Google Scholar 

  19. Miyamoto, I.: A type of uniqueness of solutions for the Dirichlet problem on a cylinder. Tohoku Math. J. 48(2), 267–292 (1996)

    Article  MathSciNet  Google Scholar 

  20. Miyamoto, I.: Harmonic functions in a cylinder which vanish on the boundary. Japan. J. Math. (N.S.) 22(2), 241–255 (1996)

    Article  MathSciNet  Google Scholar 

  21. Mizuta, Y., Shimomura, T.: Growth properties for modified Poisson integrals in a half space. Pacific J. Math. 212(2), 333–346 (2003)

    Article  MathSciNet  Google Scholar 

  22. Qiao, L.: Cylindrical Carlemans formula of subharmonic functions and its application. J. Nonlinear Sci. Appl. 11, 947–952 (2018)

    Article  MathSciNet  Google Scholar 

  23. Rădulescu, D.V., Repovs, D.D.: Partial Differential Equations with Variable Exponents. In: Variational Methods and Qualitative Analysis. Monographs and Research Notes in Mathematics. CRC Press, Boca Raton (2015)

    Google Scholar 

  24. Yu, A., Rashkovskii, L.I., Ronkin, N.: Subharmonic functions of finite order in a cone. III. Functions of completely regular growth. J. Math. Sci. 77(1), 2929–2940 (1995)

    Article  MathSciNet  Google Scholar 

  25. Ronkin, L.I.: Functions of Completely Regular Growth. Kluwer Academic Publishers Group, Dordrecht (1992)

    Book  Google Scholar 

  26. Rozenblyum, G. V., Solomyak, M. Z., Shubin, M. A.: Spectral theory of differential operators, in Results of Science and Technology, vol. 64 of Series Modern Problems of Mathematics Fundamental Directions, pp. 5–242 (1989)

  27. Siegel, D., Talvila, E.: Uniqueness for the \(n\)-dimensional half space Dirichlet problem. Pacific J. Math. 175(2), 571–587 (1996)

    Article  MathSciNet  Google Scholar 

  28. Siegel, D., Talvila, E.: Sharp growth estimates for modified Poisson integrals in a half space. Potential Anal. 15(4), 333–360 (2001)

    Article  MathSciNet  Google Scholar 

  29. Talvila, E.: Growth estimates and Phragmén-Lindelöf principles for half space problems. Ph.D. Thesis, Waterloo: University of Waterloo (1997)

  30. Yoshida, H.: Nevanlinna norm of a subharmonic function on a cone or on a cylinder. Proc. London Math. Soc. 54(2), 267–299 (1987)

    Article  MathSciNet  Google Scholar 

  31. Yoshida, H.: Harmonic majorization of a subharmonic function on a cone or on a cylinder. Pacific J. Math. 148(2), 369–395 (1991)

    Article  MathSciNet  Google Scholar 

  32. Yoshida, H., Miyamoto, I.: Solutions of the Dirichlet problem on a cone with continuous data. J. Math. Soc. Japan 50(1), 71–93 (1998)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author also thanks the anonymous referees and the editors for their helpful comments and suggestions.

Author information

Authors and Affiliations

  1. School of Mathematics and Information Science, Henan University of Economics and Law, Zhengzhou, 450046, China

    Lei Qiao

Authors
  1. Lei Qiao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lei Qiao.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Qiao, L. Cylindrical Poisson kernel method and its applications. Z. Angew. Math. Phys. 72, 176 (2021). https://doi.org/10.1007/s00033-021-01605-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-021-01605-8

Keywords

Mathematics Subject Classification

Search

Navigation