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• W1569343054 abstract "The problem of completeness of a system of elementary solutions in the space of biharmonic functions with finite energy is investigated. The problem arises during the study of infinite systems of linear algebraic equations in the asymptotic theory of plates. Actually a more general theory is developed here, including e. g. orthotropic and transversely inhomogeneous plates. The problem of existence of elementary solutions is solved at the same time. The results concerning the completeness obtained here are independent of the form of the boundary conditions at the end and can, consequently, be applied to a fairly wide class of elliptic boundary value problems which, in particular, appear in the theory of thick plates. Before the problems of completeness are discussed, we study the problem of traces for the solution of a certain elliptic equation in a semi-cylinder. The necessary and sufficient conditions are formulated for the boundary values which ensure that the solution belongs to the energy space. As we know, the stress-strain state of a plate can be separated into the internal state and the boundary layer [1–4], Construction of the boundary layer involves consecutive solutions of the plane problems of the theory of elasticity in a semistrip. Papkovich [5] and others reduce the boundary value problem of the theory of elasticity in the semi-strip x > 0, ¦y ¦⩽ 1 to finding a biharmonic Airy function, which is sought in the form u= ∑ Imσ K >0 C k ϕ k (y)e iσ k x where ϑ k are the Papkovich functions [5, 6], σ k denote the eigenvalues of a certain nonselfconjugate boundary value problem and c k are unknown constants. In this connection the author of [6] poses the problem of representing a pair of functions ƒ 1 and ƒ 2 in the form ∑ k=1 ∞ C k P k σ k =ƒ 1 , ∑ k=1 ∞ C k P k σ k =ƒ 2 where P k and Q k are differential operators determined by the boundary conditions at x = 0. Certain sufficient conditions for the uniform convergence of the series (0. 1) are given in [7, 8] for the cases when the coefficients C k can be obtained in the explicit form using the generalized conditions of orthogonality. Vorovich has shown in [9] that the present problem is related to the problem of n -tuple completeness discussed by Keldysh in [10], and suggested a novel method (realized in [11]) of investigating the expansions (0.1) based on direct study of the initial boundary value problem. In [11] the coefficients C k are uniquely defined by the boundary values of the biharmonic function and its derivatives. Thus the completeness and the basic properties of the elementary solutions are both found to be closely connected with the differential properties of the biharmonic function in a region with corners (the problem of traces). Amongst the recent investigations we note [12] where a theorem was announced concerning n -tuple completeness in the space L 2 of a part of the eigenvectors and adjoint vectors belonging to the operator bundle generated by a certain boundary problem for an elliptic equation in a semi-strip. The problem of traces for a two-dimensional region with a piecewise smooth boundary was discussed in [13], while [14] dealt with the differential properties of solutions of the general elliptic equations in regions with conical and corner points. Certain new results pertaining to the biharmonic equation are given in [15], and [16–18] deal with the behavior of the solutions of the problems of the theory of elasticity near the singular points on the boundary." @default.
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• W1569343054 date "1973-01-01" @default.
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• W1569343054 title "On the completeness of a system of elementary solutions of the biharmonic equation in a semi-strip" @default.
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• W1569343054 doi "https://doi.org/10.1016/0021-8928(73)90116-0" @default.
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