Diffraction by systems of finite strips

Abstract

The talk is devoted to wave diffraction problems for the two-dimensional Helmholtz equation. Boundary-transmission problems for a union of intervals (i.e., strips of a negligible thickness) which lie on a system of one or several parallel lines are considered in a Sobolev space setting. These problems include the mixed Dirichlet/Neumann and impedance boundary and transmission problems. Such problems for a union of intervals are reduced to equivalent systems of Wiener–Hopf equations on Lebesgue spaces over the positive real semiaxis. The matrix symbols of corresponding Wiener–Hopf operators are semialmost periodic, that is, they are continuous on the real line and have, in general, different almost periodic behavior at semineighborhoods of infinity. The solvability of obtained systems of Wiener–Hopf equations is studied on the basis of factorization techniques for oscillating and sectorial matrix functions and a corona problem approach developed in M. A. Bastos, Yu. I. Karlovich, and A. F. dos Santos [Integr. Eq. Oper. Th. 42, 22–56 (2002)], A. Bottcher, Yu. I. Karlovich, and I. M. Spitkovsky [Convolution Operators and Factorization of Almost Periodic Matrix Functions (Birkhauser-Verlag, Basel, 2002)], and L. P. Castro and F.-O. Speck [Integr. Equat. Oper. Th. 37, 169–207 (2000)].