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The Wiener-Hopf method for systems of pseudodifferential equations with an application to crack problems

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Abstract

The aim of this paper is to develop the Wiener-Hopf method for systems of pseudo-differential equations with “non-constant coefficients” and to apply it to the describtion of the asymptotic behaviour of solutions to boundary integral equations for crack problems when a crack occurs in a linear anisotropic elastic medium. The method was suggested in [15] for scalar pseudo-differential equations with “constant coefficients” and applied in [7] to the crack problems in the isotropic case. The existence and a-priori smoothness of solutions for the anisotropic case has been proved in [11, 12], while the isotropic case has been treated earlier in [7, 25, 41, 50]. Our results improve even those for the isotropic case obtained in [7, 50]. Asymptotic estimates for the behaviour of solutions in the anisotropic case have been obtained in [28] by a different method.

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Authors and Affiliations

  1. Tbilisi Mathematical Institute, Z. Rukhadze str. 1, 380093, Tbilisi, Rep. of Georgia

    R. Duduchava

  2. Mathematisches Institut A, Universität Stuttgart, D-70550, Stuttgart, Fed. Rep. Germany

    W. L. Wendland

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  1. R. Duduchava
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  2. W. L. Wendland
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In memoriam, dedicated to Professor Dr. V.D. Kupradze on the occasion of the 90th anniversary of his birth

This work was carried out during the first author's visit in Stuttgart in 1992 and supported by the DFG priority research programme “Boundary Element Methods” within the guest-programme We-659/19-2.

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Duduchava, R., Wendland, W.L. The Wiener-Hopf method for systems of pseudodifferential equations with an application to crack problems. Integr equ oper theory 23, 294–335 (1995). https://doi.org/10.1007/BF01198487

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