About Symmetry in Partially Hinged Composite Plates

Abstract

We consider a partially hinged composite plate problem and we investigate qualitative properties, e.g. symmetry and monotonicity, of the eigenfunction corresponding to the density minimizing the first eigenvalue. The analysis is performed by showing related properties of the Green function of the operator and by applying polarization with respect to a fixed plane. As a by-product of the study, we obtain a Hopf type boundary lemma for the operator having its own theoretical interest. The statements are complemented by numerical results.

Elasticity solution for a 3D hollow cylinder axially loaded at the end faces

“…Recently some mathematical models for suspension bridges [11] have been developed with the scope to understand and, then, to prevent instability phenomena; they were studied models for suspension bridges with geometrical nonlinearities, e.g. see [7,8], models for partially hinged plates, see [9], models for non homogeneous partially hinged plates, see [1,2,3], models for homogeneous beams with intermediate piers [10] and non homogeneous beams [4]. In all these cases the application of analytical methods to real problems allowed to find suggestions and practical remedies that can be discussed with engineers.…”

“…Given Ω ⊂ R 3 an elastic homogeneous solid body, we denote by u : Ω → R 3 the displacement vector at any point of the reference configuration of the elastic body itself, see the list of notations at the end of the paper. We denote by Tu the stress tensor and by λ and µ the classical Lamé constants; it is known that λ and µ may be expressed in terms of the Young modulus E and Poisson ratio ν ∈ (−1, 1 2 ) as…”

Spectral optimization of inhomogeneous plates

“…The study of eigenmodes optimisation is central to the theory of inhomogeneous elastic plates and is of great applicative relevance. A vast literature has been devoted to the analysis of spectral optimisation problems for biharmonic operators, modelling plates of varying density and thickness under different settings [3,4,6,8,9,10,11,14,18,19]. In addition, several contributions are devoted to inverse problems arising in the study of such inhomogeneous plates [17,25,26].…”

“…The most general formulation of the optimisation problem under consideration, covering questions from [3,4,6,14,18,19], is the study of the qualitative properties of solutions to the minimisation problem inf D,g Λ(D, g).…”

“…In these contributions, rearrangements arguments and Talenti inequalities are used in order to derive Faber-Krahn-like inequalities, delivering information on the geometry of minimizers. On the other hand, the optimisation of the thickness D is mostly treated numerically [6,17,25,26] and the existence of a minimizer D * is usually not ascertained, to the best of our knowledge. Let us stress that existence in this setting can be quite delicate to obtain.…”