Completeness of root functions for thermal conduction in a strip with piecewise continuous coefficients

We consider the boundary layer problem associated with the steady thermal conduction problem in a thin laminated plate. Two cases of boundary conditions, Dirichlet and Neumann, are treated in the paper. Transmission conditions across the interfaces should be added since the plate is laminated. The study of the structure of the solution in the matching region of the layer with the basis solution in the plate leads to consideration of an eigenvalue problem for a second-order operator pencil with piecewise continuous coefficients and the corresponding boundary and transmission conditions. Twofold completeness of root functions of the latter problem is proved. The boundary layer term can then be expressed as a combination of these functions.