On the Behavior of Clamped Plates under Large Compression

Abstract

We determine the asymptotic behaviour of eigenvalues of clamped plates under large compression, by relating this problem to eigenvalues of the Laplacian with Robin boundary conditions. Using the method of fundamental solutions, we then carry out a numerical study of the extremal domains for the first eigenvalue, from which we see that these depend on the value of the compression, and start developing a boundary structure as this parameter is increased. The corresponding number of nodal domains of the first eigenfunction of the extremal domain also increases with the compression.2010 Mathematics Subject Classification. Primary 35J30. Secondary 35P15, 35P20, 49R50, 74K20.

Positivity for the clamped plate equation under high tension

“…Hence the existing literature for these eigenvalue problems is quite vast and is still developing, see, e.g. [8,4] and the references therein.…”

The Bilaplacian with Robin boundary conditions

“…Here α ∈ R, where α > 0 corresponds to a plate under tension, while for α < 0 it is under compression instead. The dependence on the tension parameter α has been considered in various contexts, both for the analysis of the behaviour of the eigenvalues and focusing on limiting regimes, see [1,23,19,21,28,29,54]. We note also that this perturbation arises naturally in the context and study of buckling phenomena for plates, where α becomes a so-called eigenvalue of the operator pencil (4) ∆ 2 u = −Λ∆u.…”

“…Similarly to problem (1), when d = 2 the equation ( 5) models the behaviour of a three-dimensional thin plate of cross-section Ω subject to the load f , and the solution u represents the displacement of the section of the plate with respect to its position at rest. The constants α, β, γ, σ are mechanical constants related to the response of the material with respect to mechanical stimulations; in particular, as mentioned, the Poisson ratio σ measures the stiffness of the material, and α is the ratio of tension to flexural rigidity.…”

“…First and foremost, we introduce a bilinear form (see (6)) as the natural fourth-order analogue of the form associated with problem (1); in turn, this form indeed has problem (5) as its strong formulation. In particular, we show that problem (5) is indeed associated with a self-adjoint operator on L 2 (Ω) with compact resolvent, for any possible choice of the parameters (Theorem 2.1).…”

Positivity for the clamped plate equation under high tension

“…Hence the existing literature for these eigenvalue problems is quite vast and is still developing, see e.g. [BuoKe21], [AnBuoFr19] and the references therein. Other modifications for (1.1) concerning positivity issues are e.g.…”