Some degenerate elliptic systems and applications to cusped plates

Abstract

The tension-compression vibration of an elastic cusped plate is studied under all the reasonable boundary conditions at the cusped edge, while at the noncusped edge displacements and at the upper and lower faces of the plate stresses are given.Mathematics Subject Classification 2000. Primary 74K20; Secondary 35J70

Existence and uniqueness theorems for cusped prismatic shells in the Nth hierarchical model

“…In the case of the N = 1 approximation, the partial differential equations for the unknown vector v = (v 10 , v 20 , v 30 The corresponding bilinear and quadratic forms are 2 31 } d In accordance with the results in Sections 3 and 4, the variational problem…”

“…The partial differential equations for the N = 2 approximation for the unknown vector v = (v 10 , v 20 , v 30 , v 11 , v 21 , v 31 , v 12 , v 22 , v 32 ) and the corresponding bilinear and quadratic forms can be expressed explicitly with the help of (21) and (25). Again, with the help of the results obtained in Sections 3 and 4 we conclude that the variational problem B (2) (v, v * ) = F, v * for all v * ∈ X 2 (69) APPENDIX A Let be as in Section 2 and let D( ) be a space of infinitely differentiable functions with compact support in .…”

“…More details can be found in a survey paper of Dauge et al [29]. Jaiani and Schulze [30] considered the tension-compression vibration of cusped plates was in terms of the N = 1 approximation of I. Vekua's hierarchical models. For the corresponding degenerate static system when the thickness is given by 2h = x 2 , = const.>0, x 2 0, the homogeneous Dirichlet problem was studied in Devdariani [31] (see also [32]).…”

“…<1, then the components v10 , v20 , and v30 have the zero traces on the whole of the boundary * , while the components v 11 , v 21 , and v 31 have the zero traces on the part 1 ⊂ * and, in general, they have no traces on the part 0 ⊂ * due to order degeneration of Equations (64).DOI: 10.1002/mma EXISTENCE AND UNIQUENESS THEOREMS FOR CUSPED PRISMATIC SHELLS 1363 By Theorem 5.1 for <1 and = 1 3 , we have the following embeddings: ( )] 3 = Y 1 and the corresponding norms are equivalent, i.e. actually the sets of vector functions Y 1 and X 1 coincide.…”

On Some Bending Problems of Prismatic Shell with the Thickness Vanishing at Infinity

“…Works of Babuska, Gordeziani, Guliaev, Khoma, Khvoles, Meunargia, Schwab, Vashakmadze, Zhgenti, Jaiani, Tsiskarishvili, M. and G. Avalishvili, Wendland, Natroshvili, Kharibegashvili, Chinchaladze, Gilbert, and others are devoted to further analysis of I.Vekua's models (rigorous estimation of the modeling error, numerical solutions, etc.) and their generalizations (see, e.g., [2], [3], [4], [5], [6], [7], [8], [9], [12], [16] …”