Singular Localised Boundary-Domain Integral Equations of Acoustic Scattering by Inhomogeneous Anisotropic Obstacle

We consider the time-harmonic acoustic wave scattering by a bounded {\it anisotropic inhomogeneity} embedded in an unbounded {\it 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 {\it singular localised boundary-domain integral equations}. Fredholm properties of the corresponding {\it localised boundary-domain integral operator} are studied and its invertibility is established in appropriate Sobolev-Slobodetskii and Bessel potential spaces, which implies existence and uniqueness results for the localised boundary-domain integral equations system and the corresponding acoustic scattering transmission problem.