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


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.


Boundary Condition Dirichlet Boundary Zeta Function Dirichlet Boundary Condition Elastic Plate 
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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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