The Wiener-Hopf method for systems of pseudodifferential equations with an application to crack problems.
The paper presents general results on Fredholmness and solvability in Bessel potential and Besov spaces for systems of pseudodifferential equations on a compact domain with smooth boundary, based on a refined Wiener-Hopf factorization technique. This developes the method suggested by \textit{G. Eskin} [Boundary value problem for elliptic pseudodifferential equations, AMS Translations of Mathematical Monographs 52 (1981; Zbl 0458.35002)] for scalar pseudodifferential equations with ``constant coefficients''. Furthermore, the Wiener-Hopf method is applied to the boundary integral equations corresponding to the Dirichlet and Neumann problems for an anisotropic medium, yielding short proofs for solvability and regularity results by Duduchava, Natroshvili and Shargorodsky. Finally, new results on the asymptotic behaviour of solutions to crack problems are given, which improve even those for the isotropic case obtained by \textit{M. Costabel} and \textit{E. P. Stephan} [Integral Equations Oper. Theory 10, 467--504 (1987; Zbl 0632.73091)].
scientific article; zbMATH DE number 821683
scientific article