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A green's function for the annulus

Abstract

In this paper we find an expression for Green's junction for the operator Δ2 in 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.

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Sponsored by the «Civilingenjör Gustaf Sigurd Magnusons fond för främjande av vetenskapen inom ämnet matematik» of the Royal Swedish Academy of Sciences (Kungl. Vetenskapsakademien) and also in part by GA AV ČR grant No. 119106.

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Engliš, M., Peetre, J. A green's function for the annulus. Annali di Matematica pura ed applicata 171, 313–377 (1996). https://doi.org/10.1007/BF01759391

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Keywords

  • Boundary Condition
  • Dirichlet Boundary
  • Zeta Function
  • Dirichlet Boundary Condition
  • Elastic Plate