Singular localised boundary-domain integral equations of acoustic scattering by inhomogeneous anisotropic obstacle

Abstract

We consider the time-harmonic acoustic wave scattering by a bounded anisotropic inhomogeneity embedded in an unbounded anisotropic homogeneous medium. The material parameters may have discontinuities across the interface between the inhomogeneous interior and homogeneous exterior regions. The corresponding mathematical problem is formulated as a transmission problems for a second-order elliptic partial differential equation of Helmholtz type with discontinuous variable coefficients. Using a localised quasi-parametrix based on the harmonic fundamental solution, the transmission problem for arbitrary values of the frequency parameter is reduced equivalently to a system of singular localised boundary-domain integral equations. Fredholm properties of the corresponding localised boundary-domain integral operator are studied and its invertibility is established in appropriate Sobolev-Slobodetskii (Bessel potential) spaces, which implies existence and uniqueness results for the localised boundary-domain integral equations system and the corresponding acoustic scattering transmission problem.