On Some Initial-Boundary Transmission Problems.

A general approach to transmission problems was considered in the author’s previous work ([14], [15]). It consists in the fact that the solution of an inhomogeneous problem is sought in the form of a sum of solutions of auxiliary homogeneous problems. In these auxiliary problems, the inhomogeneity is contained only in one place, that is, either in the equation or in the boundary condition. The solution of each of the auxiliary problems is found by means of the corresponding Green’s formulas ([10], [14]). The solution of the original problem is the sum of the solutions of the auxiliary problems. This general scheme is applied to various configurations of Lipschitz domains with Lipschitz boundaries. Theorems on the existence and uniqueness of a weak solution for each problem are obtained.
The approach described above is also applied to spectral conjugation problems for one, two, and three adjacent regions. As a result of studying these problems, the same operator bundle is obtained. It was investigated by the methods of the spectral theory of operator pencils ([9]).
The general scheme is used for initial-boundary value problems that generate the spectral in this paper. The derivatives with respect to time enter not only into the equation, but also into the boundary conditions in these problems. Four problems for one domain are considered, theorems on the existence and uniqueness of a strong solution with values in the corresponding Hilbert space are obtained. Similar problems for two and three adjacent regions are also studied. The equations satisfied by their solutions are reduced to the same Cauchy problems as in the case of one region. Therefore, the same theorems on existence and uniqueness are valid for them.

Keywords: Hilbert space, strong solution, basicity, asymptotics, transmission problem, initial- boundary problem, operator pencil.