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 homogeneousmedium. The materialparametersmay have discontinuitiesacross the interfacebetween the inhomogeneous interior and homogeneous exterior regions. The corresponding mathematicalproblemis formulatedas a transmissionproblemsfora secondorderelliptic 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 boundarydomainintegral operatorarestudiedanditsinvertibilityisestablishedinappropriate Sobolev-Slobodetskii and Bessel potential spaces, which implies existence and uniquenessresults for the localised boundary-domainintegralequationssystem and the correspondingacoustic scattering transmission problem.