O. Chkadua, S. E. Mikhailov, and D. Natroshvili
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.
Mathematics Subject Classification: 35J25, 31B10, 45P05, 45A05, 47G10, 47G30, 47G40
Key words and phrases: Partial differential equation, transmission problem, interface crack problem, mixed problem, localized parametrix, localized boundary-domain integral equations, pseudo-differential equation