Solution of Irregular Systems of Partial Differential Equations Using Skeleton Decomposition of Linear Operators

Abstract

The linear system of partial dierential 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 dierential operators of such system are assumed to have suciently smooth coecients. In the concrete situations the domains of such dierential 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 nite 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 dierential 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 dierential equations summarized in the monographs MR 87a:58036, Zbl 1027.47001.

Solvability of Nonlinear Equations in Case of Branching Solutions

“…. (17) into (15) gives the following branching LS system For symmetry let us put λ = ξ n+1 in (18). Let L(ξ 1 , .…”

“…. , ξ n+1 ) be one of the left hand sides of system (18). Eliminate on the corresponding power ξ i for sake of clarity we assume L(ξ 1 , .…”

“…Here rank[a jk ] k=1,...,n+1, j=1,...,n = n. Then branching equation (18) call quasilinear. In this case it is easy to constract the asymptotic of solution for eq.…”

“…Let conditions 1 and 2 are fulfilled for branching system (18) and ξ 0 is solution of system (20) is full rank solution. Let r = min(α 1 , .…”

“…Trenogin [4] the basement of the analytical theory of branching solutions in Banach spaces with applications is given. These works contributed to the modern functional analysis development with many applications [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22].…”