Localized direct segregated boundary-domain integral equations for variable-coefficient transmission problems with interface crack

Abstract

Some transmission problems for scalar second order elliptic partial differential equations are considered in a bounded composite domain consisting of adjacent anisotropic subdomains having a common interface surface. The matrix of coefficients of the differential operator has a jump across the interface but in each of the adjacent subdomains is represented as the product of a constant matrix by a smooth variable scalar function. The Dirichlet or mixed type boundary conditions are prescribed on the exterior boundary of the composite domain, the Neumann conditions on the the interface crack surfaces and the transmission conditions on the rest of the interface. Employing the parametrix-based localized potential method, the transmission problems are reduced to the localized boundary-domain integral equations. The corresponding localized boundary-domain integral operators are investigated and their invertibility in appropriate function spaces is proved.