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Annali di Matematica Pura ed Applicata

, Volume 171, Issue 1, pp 313–377 | Cite as

A green's function for the annulus

  • Miroslav Engliš
  • Jaak Peetre
Article

Abstract

In this paper we find an expression for Green's junction for the operator Δ2in a planar circular annulus with Dirichlet boundary conditions (clamped elastic plate). We likewise determine the corresponding Poisson type kernels and the harmonic Bergman kernel. These results come in terms of certain new transcendental functions which in a natural way generalize the Weierstrass zeta function. They are analogous to the result of R.Courant D.Hubert (Methoden der Mathematischen Physik I (3. Aufl.), Springer-Verlag, Berlin, Heidelberg, New York (1968), pp. 335-337)and H.Villat (Rend. Circ. Mat. Palermo,33 (1912), pp. 134–175)respectively. As an application we show that, regardless of the size of the ratio of the radii of the bounding circles, the Green's function always assumes negative values, which constitutes another rather striking counter-example to the wellknown Boggio-Hadamard conjecture.

Keywords

Boundary Condition Dirichlet Boundary Zeta Function Dirichlet Boundary Condition Elastic Plate 
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References

  1. [1]
    E. Almansi,Sulle integrazione dell'equazione differenziale Δ 2n=0, Ann. Mat. Pura Appl.,2 (1898), pp. 1–51.Google Scholar
  2. [2]
    M.Ashbaugh, Electronic mail, May17, 1994.Google Scholar
  3. [3]
    M.Ashbaugh, Electronic mail, March2, 1995.Google Scholar
  4. [4]
    N. Aronszajn -Th. M. Creese -L. J. Lipkin,Polyharmonic Functions, Clarendon Press, Oxford (1983).Google Scholar
  5. [5]
    B.Bojarski (Boyarskii),Remarks on polyharmonic functions operators and conformal mappings in space, inTrudy Vsesoyuznogo Simposiuma v Tbilisi 21–23 aprelya 1982 g., pp. 49–56 (Russian).Google Scholar
  6. [6]
    C. V. Coffman -R. J. Duffin,On the fundamental eigenfunctions of a clamped punctured disc, Adv. Appl. Math.,13 (1992), pp. 142–151.Google Scholar
  7. [7]
    C. V. Coffman -R. J. Duffin -D. H. Shaffer,The fundamental mode of vibration of a clamped annular plate is not of one sign, inC. V. Duffin -R. J. Shaffer (eds.),Constructive Approaches to Mathematical Models, pp. 267–277, Academic Press, New York (1979).Google Scholar
  8. [8]
    R. Courant -D. Hilbert,Methoden der Mathematischen Physik I, 3. Aufl., Springer-Verlag, Berlin-Heidelberg-New York (1968).Google Scholar
  9. [9]
    R. J. Duffin,On a question of Hadamard concerning super-biharmonic functions, J. Math. Phys.,27 (1949), pp. 253–258.Google Scholar
  10. [10]
    C. V. Duffin -R. J. Shaffer,On the modes of vibration of a ring-shaped plate, Bull. Am. Math. Soc.,58 (1952), p. 652.Google Scholar
  11. [11]
    M.Englis - J.Peetre,Covariant differential operators and Green functions, Ann. Polon. Math., to appear.Google Scholar
  12. [12]
    P. R. Garabedian,A partial differential equation arising in conformai mapping, Pac. J. Math.,1 (1951), pp. 485–523.Google Scholar
  13. [13]
    W. K. Hayman -B. Korenblum,Representation and uniqueness of polyharmonic functions, J. Anal. Math.,60 (1993), pp. 113–133.Google Scholar
  14. [14]
    P. J. H. Hedenmalm,A computation of Green functions for the weighted biharmonic operators \(\Delta |z|^{ - 2\alpha } \Delta \),with α > −1}, Duke Math. J.,75 (1994), pp. 51–78.Google Scholar
  15. [15]
    S. Janson -J. Peetre,Harmonic interpolation, inInterpolation Spaces and Allied Topics in Analysis, Proceedings,Lund,1983, Lecture Notes in Mathematics,1070, pp. 92–124, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo (1984).Google Scholar
  16. [16]
    G. Kowalewsky,Einführung in die Determinantentheorie einsckliessend der unendlichen und der Fredholmschen Determinanten, Veit, Leipzig (1909).Google Scholar
  17. [17]
    J. Peetre,Orthogonal polynomials arising in connection with Hankel forms of higher weight, Bull. Sci. Math. (2),116 (1992), pp. 265–284.Google Scholar
  18. [18]
    I. Schur,Über Potenzreihen, die im Innern des Einheitskreises beschränkt sind, J. Reine Angew. Math.,147 (1917), pp. 205–232; English translation: I.Gohberg (editor),I. Schur methods in operator theory and signal processing, pp. 31–88 (Operator Theory: Advances and Applications, vol.18), Birkhäuser, Basel (1986).Google Scholar
  19. [19]
    G. Szegö,Collected Papers, Vol.3, 1945–1972 (edited byRichard Askey), Birkhäuser, Basel-Boston-Stuttgart (1982).Google Scholar
  20. [20]
    O. Venske,Zur Integration der Gleichung ΔΔu=0für ebene Bereiche, Nachr. K. Gesell. Wiss. Göttingen No. 1 (1890), pp. 27–33.Google Scholar
  21. [21]
    H. Villat,Le problème de Dinchlet dans une aire annulaire, Rend. Circ. Mat. Palermo,33 (1912), pp. 134–175.Google Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata 1996

Authors and Affiliations

  • Miroslav Engliš
    • 1
  • Jaak Peetre
    • 2
  1. 1.Mathematical InstituteAcademy of SciencesPrague 1Czech Republic
  2. 2.Department of MathematicsLund UniversityLundSweden

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