Skip to main content
SpringerLink
Log in

The Parqueting-Reflection Principle

  • Conference paper
Current Trends in Analysis and Its Applications

Part of the book series: Trends in Mathematics ((RESPERSP))

Abstract

For certain plane domains with boundaries composed by arcs from circles and straight lines the parqueting-reflection principle is used to construct the Schwarz, Green, and Neumann kernels for solving the Schwarz, Dirichlet, and Neumann boundary value problems for the inhomogeneous Cauchy–Riemann and the Poisson equation, respectively.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. M.S. Akel, H.S. Hussein, Two basic boundary-value problems for the inhomogeneous Cauchy–Riemann equation in an infinite sector. Adv. Pure Appl. Math. 3, 315–328 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  2. H. Begehr, Boundary value problems in complex analysis, I, II. Bol. Asoc. Mat. Venez. XII, 65–85 (2005), 217–250

    MathSciNet  Google Scholar 

  3. H. Begehr, T. Vaitekhovich, Polyharmonic Green functions for particular plane domains, in Proc. 6. Congress of Romanian Mathematicians, Bucharest, Romania, 2007, ed. by L. Beznea et al.. Romanian Acad. Sci., vol. 1 (Publ. House, Bucharest, 2009), pp. 119–126

    Google Scholar 

  4. H. Begehr, T. Vaitekhovich, Harmonic boundary value problems in half disc and half ring. Funct. Approx. 40, 251–282 (2009). Part 2

    Article  MATH  MathSciNet  Google Scholar 

  5. H. Begehr, T. Vaitekhovich, Green functions, reflections, and plane parqueting. Eurasian Math. J. 1, 17–31 (2010)

    MATH  MathSciNet  Google Scholar 

  6. H. Begehr, T. Vaitekhovich, Green functions, reflections, and plane parqueting, revisited. Eurasian Math. J. 2, 139–142 (2011)

    MATH  MathSciNet  Google Scholar 

  7. H. Begehr, T. Vaitekhovich, How to find harmonic Green functions in the plane. Complex Var. Elliptic Equ. 56, 1169–1181 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  8. H. Begehr, T. Vaitekhovich, Harmonic Dirichlet problem for some equilateral triangle. Complex Var. Elliptic Equ. 57, 185–196 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  9. H. Begehr, T. Vaitekhovich, The Parqueting-Reflection Principle for Constructing Green Functions, ed. by S.V. Rogosin, M.V. Dubatovskaya. Analytic Methods of Analysis and Differential Equations: AMADE 2012 (Cambridge Scientific Publishers, Cottenham, 2013), pp. 11–20

    Google Scholar 

  10. H. Begehr, T. Vaitekhovich, Schwarz problem in lens and lune. Complex Var. Elliptic Equ. 59, 76–84 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  11. E. Gaertner, Basic complex boundary value problems in the upper half plane. Ph.D. thesis, FU Berlin (2006). www.diss.fu-berlin.de/diss/receive/FUDISS_thesis_000000002129

  12. B. Shupeyeva, Harmonic boundary value problems in a quarter ring domain. Adv. Pure Appl. Math. 3, 393–419 (2012); 4, 103–105 (2013)

    MATH  MathSciNet  Google Scholar 

  13. B. Shupeyeva, Somme basic boundary value problems for complex partial differential equations in quarter ring and half hexagon. Ph.D. thesis, FU Berlin (2013). www.diss.fu-berlin.de/diss/receive/FUDISS_thesis_000000094596

  14. B. Shupeyeva, Boundary value problems and method of reflection for quarter ring and half hexagon, in Current Trends in Analysis and its Applications, Trends in Mathematics, ed. by V.V. Mityushev, W. Ruzhansky (Springer Switzerland, Basel, 2015), pp. 59–66

    Google Scholar 

  15. T. Vaitsiakhovich, Boundary value problems for complex partial differential equations in a ring domain. Ph.D. thesis, FU Berlin (2008). www.diss.fu-berlin.de/diss/receive/FUDISS_thesis_000000003859

  16. Y. Wang, Boundary value problems for complex partial differential equations in fan-shaped domains. Ph.D. thesis, FU Berlin (2011). www.diss.fu-berlin.de/diss/receive/FUDISS_thesis_000000021359

  17. Y.-F. Wang, Y.-J. Wang, Schwarz-type problem of nonhomogeneous Cauchy–Riemann equation on a triangle. J. Math. Anal. Appl. 377(2), 557–570 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  18. Y. Wang, J. Du, Harmonic Dirichlet problem in a ring sector, in Current Trends in Analysis and its Applications, Trends in Mathematics, ed. by V.V. Mityushev, W. Ruzhansky (Springer Switzerland, Basel, 2015), pp. 67–75

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Math. Inst. FU Berlin, Arnimallee 3, 14195, Berlin, Germany

    Heinrich Begehr

Authors
  1. Heinrich Begehr
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Heinrich Begehr .

Editor information

Editors and Affiliations

  1. Dept. of Computer Sc. & Comp. Methods, Pedagogical University, Kraków, Poland

    Vladimir V. Mityushev

  2. Imperial College London, London, United Kingdom

    Michael V. Ruzhansky

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Begehr, H. (2015). The Parqueting-Reflection Principle. In: Mityushev, V., Ruzhansky, M. (eds) Current Trends in Analysis and Its Applications. Trends in Mathematics(). Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-12577-0_11

Download citation

Publish with us

Policies and ethics

Search

Navigation