Harmonic vibration of prismatic shells in zero approximation of Vekua's hierarchical models

Abstract

In this article elastic cusped symmetric prismatic shells (i.e., plates of variable thickness with cusped edges) in the zero approximation of I.Vekua's hierarchical models is considered. The well-posedness of the boundary value problems (BVPs) under the reasonable boundary conditions at the cusped edge and given displacements at the non-cusped edge is studied in the case of harmonic vibration. The approach works also for non-symmetric prismatic shells word for word. The classical and weak setting of the BVPs in the case of the zero approximation of hierarchical models is considered. Appropriate weighted functional spaces are introduced. Uniqueness and existence results for the variational problem are proved. The structure of the constructed weighted space is described and its connection with weighted Sobolev spaces is established. Moreover, some sufficient conditions for a linear functional arising in the right-hand side of the variational equation to be bounded are given.

Keywords:

• cusped plates
• cusped prismatic shells
• degenerate elliptic systems
• weighted spaces
• Hardy's inequality
• Korn's weighted inequality

AMS Subject Classifications::

• 74K20
• 35J70

Acknowledgements

The authors are very grateful to Prof. G. Jaiani, Prof. S. Kharibegashvili and Prof. D. Natroshvili for helpful discussions.