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Solution of irregular systems of partial differential equations using skeleton decomposition of linear operators. (English) Zbl 1378.35332

Summary: The linear system of partial differential equations is considered. It is assumed that there is an irreversible linear operator in the main part of the system. The operator is assumed to enjoy the skeletal decomposition. The differential operators of such system are assumed to have sufficiently smooth coefficients. In the concrete situations the domains of such differential operators are linear manifolds of smooth enough functions with values in Banach space. Such functions are assumed to satisfy additional boundary conditions. The concept of a skeleton chain of linear operator is introduced. It is assumed that the operator generates a skeleton chain of the finite length. In this case, the problem of solution of a given system is reduced to a regular split system of equations. The system is resolved with respect to the highest differential expressions taking into account certain initial and boundary conditions. The proposed approach can be generalized and applied to the boundary value problems in the nonlinear case. Presented results develop the theory of degenerate differential equations summarized in the monographs [N. A. Sidorov, General regularization questions in problems of bifurcation theory. (Obshchie voprosy regulyarizatsii v zadachakh teorii vetvleniya) (Russian). Irkutsk: Izdatel’stvo Irkutskogo Universiteta (1982; Zbl 0703.58002); N. Sidorov et al., Lyapunov-Schmidt methods in nonlinear analysis and applications. Dordrecht: Kluwer Academic Publishers (2002; Zbl 1027.47001)].

MSC:

35R25 Ill-posed problems for PDEs
35G15 Boundary value problems for linear higher-order PDEs
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