We show that, in contrast to the case of Laplace eigenfunctions, the nodal set of high energy eigenfunctions of the clamped plate problem is not necessarily dense, and can in fact exhibit macroscopic "nodal voids". Specifically, we show that there are small deformations of the unit disk admitting a clamped plate eigenfunction of arbitrarily high frequency that does not vanish in a disk of radius 0.44.
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.
In this paper, we study eigenvalues of a clamped plate problem. We obtain a lower bound for eigenvalues, which gives an important improvement of results due to Levine and Protter.
In this article we consider positivity issues for the clamped plate equation with high tension $γ>0$. This equation is given by $Δ^2u - γΔu=f$ under clamped boundary conditions. Here we show, that given a positive $f$, i.e. upwards pushing, we find a $γ_0>0$ such that for all $γ\geq γ_0$ the bending $u$ is indeed positive. This $γ_0$ only depends on the domain and the ratio of the $L^1$ and $L^\infty$ norm of $f$. In contrast to a recent result by Cassani&Tarsia, our approach is valid in all dimensions.
We formulate the minimization of the buckling load of a clamped plate as a free boundary value problem with a penalization term for the volume constraint. As the penalization parameter becomes small we show that the optimal shape problem with prescribed volume is solved. In addition, we discuss two different choices for the penalization term.
This paper studies eigenvalues of the clamped plate problem on a bounded domain in an $n$-dimensional Euclidean space. We give an estimate for the gap between $\sqrt {Γ_{k+1}-Γ_{1}}$ and $\sqrt {Γ_{k}-Γ_{1}}$, for any positive integer $k$. According to the asymptotic formula of Agmon and Pleijel, we know, the gap between $\sqrt {Γ_{k+1}-Γ_{1}}$ and $\sqrt {Γ_{k}-Γ_{1}}$ is bounded by a term with a lower order $k^{\frac1n}$ in the sense of the asymptotic formula of Agmon and Peijel, where $Γ_j$ denotes the $j^{^{\text{th}}}$ eigenvalue of the clamped plate problem.
We prove the existence of an optimal domain for minimizing the buckling load among all, possibly unbounded, open subsets of $\mathbb{R}^n$ ($n\geq 2$) with given measure. Our approach is based on the extension of a 2-dimensional existence result of Ashbaugh and Bucur and on the idea of Alt and Caffarelli to focus on the eigenfunction.
In the 19th century, Lord Rayleigh conjectured that among all clamped plates with given area, the disk minimizes the fundamental tone. In the 1990s, N. S. Nadirashvili proved the conjecture in $\mathbb{R}^2$ and M. S. Ashbaugh und R. D. Benguria gave a proof in $\mathbb{R}^2$ and $\mathbb{R}^3$. In the present paper, we prove existence of an optimal domain for minimizing the fundamental tone among all open and bounded subsets of $\mathbb{R}^n$, $n\geq 4$, with given measure. We formulate the minimization of the fundamental tone of a clamped plate as a free boundary value problem with a penaliz…
We affirmatively solve the analogue of Lord Rayleigh's conjecture on Riemannian manifolds with positive Ricci curvature for any clamped plates in 2 and 3 dimensions, and for sufficiently large clamped plates in dimensions beyond 3. These results complement those from the flat (M. Ashbaugh & R. Benguria, 1995, and N. Nadirashvili, 1995) and negatively curved (A. Kristály, 2020) cases that are valid only in 2 and 3 dimensions, and at the same time also provide the first positive answer to Lord Rayleigh's conjecture in higher dimensions. The proofs rely on an Ashbaugh-Benguria-Nadirashvili-Talent…
In this paper we investigate the limit behavior of the solution to quasi-static Biot's equations in thin poroelastic plates as the thickness tends to zero. We choose Terzaghi's time corresponding to the plate thickness and obtain the strong convergence of the three-dimensional solid displacement, fluid pressure and total poroelastic stress to the solution of the new class of plate equations. In the new equations the in-plane stretching is described by the 2D Navier's linear elasticity equations, with elastic moduli depending on Gassmann's and Biot's coefficients. The bending equation is couple…
$\mathfrak{L}_{II}$ operator is introduced by Y.-L. Xin (\emph{Calculus of Variations and Partial Differential Equations. 2015, \textbf{54}(2):1995-2016)}, which is an important extrinsic elliptic differential operator of divergence type and has profound geometric meaning. In this paper, we extend $\mathfrak{L}_{II}$ operator to more general elliptic differential operator $\mathfrak{L}_ν$, and investigate the clamped plate problem of bi-$\mathfrak{L}_ν$ operator, which is denoted by $\mathfrak{L}_ν^{2}$, on the complete Riemannian manifolds. A general formula of eigenvalues for the $\mathfrak{…
We study Lord Rayleigh's problem for clamped plates on an arbitrary $n$-dimensional $(n\geq 2)$ Cartan-Hadamard manifold $(M,g)$ with sectional curvature $\textbf{K}\leq -κ^2$ for some $κ\geq 0.$ We first prove a McKean-type spectral gap estimate, i.e. the fundamental tone of any domain in $(M,g)$ is universally bounded from below by $\frac{(n-1)^4}{16}κ^4$ whenever the $κ$-Cartan-Hadamard conjecture holds on $(M,g)$, e.g. in 2- and 3-dimensions due to Bol (1941) and Kleiner (1992), respectively. In 2- and 3-dimensions we prove sharp isoperimetric inequalities for sufficiently small clamped pl…
We address the long-time behavior of a non-rotational von Karman plate in an inviscid potential flow. The model arises in aeroelasticity and models the interaction between a thin, nonlinear panel and a flow of gas in which it is immersed [6, 21, 23]. Recent results in [16, 18] show that the plate component of the dynamics (in the presence of a physical plate nonlinearity) converge to a global compact attracting set of finite dimension; these results were obtained in the absence of mechanical damping of any type. Here we show that, by incorporating mechanical damping the full flow-plate system,…
In this paper we propose two numerical algorithms to derive the extremal principal eigenvalue of the bi-Laplacian operator under Navier boundary conditions or Dirichlet boundary conditions. Consider a non-homogeneous hinged or clamped plate $Ω$, the algorithms converge to the density functions on $Ω$ which they yield the maximum or minimum basic frequency of the plate.
We consider the buckling eigenvalue problem for a clamped plate in the annulus. We identify the first eigenvalue in dependence of the inner radius, and study the number of nodal domains of the corresponding eigenfunctions. Moreover, in order to investigate the asymptotic behavior of eigenvalues and eigenfunctions as the inner radius approaches the outer one, we provide an analytical study of the buckling problem in rectangles with mixed boundary conditions.
We consider the inverse problem of determining the Winkler subgrade reaction coefficient of a slab foundation modelled as a thin elastic plate clamped at the boundary. The plate is loaded by a concentrated force and its transversal deflection is measured at the interior points. We prove a global Hölder stability estimate under (mild) regularity assumptions on the unknown coefficient.
In the present article, we consider a thermoelastic plate of Reissner-Mindlin-Timoshenko type with the hyperbolic heat conduction arising from Cattaneo's law. In the absense of any additional mechanical dissipations, the system is often not even strongly stable unless restricted to the rotationally symmetric case, etc. We present a well-posedness result for the linear problem under general mixed boundary conditions for the elastic and thermal parts. For the case of a clamped, thermally isolated plate, we show an exponential energy decay rate under a full damping for all elastic variables. Rest…
We consider the dynamics of an electrostatically actuated thin elastic plate being clamped at its boundary above a rigid plate. The model includes the harmonic electrostatic potential in the three-dimensional time-varying region between the plates along with a fourth-order semilinear parabolic equation for the elastic plate deflection which is coupled to the square of the gradient trace of the electrostatic potential on this plate. The strength of the coupling is tuned by a parameter $λ$ proportional to the square of the applied voltage. We prove that this free boundary problem is locally well…
The aim of this article is to provide a simple and unified way to obtain the sharp upper bounds of nodal sets of eigenfunctions for different types of eigenvalue problems on real analytic domains. The examples include biharmonic Steklov eigenvalue problems, buckling eigenvalue problems and champed-plate eigenvalue problems. The geometric measure of nodal sets are derived from doubling inequalities and growth estimates for eigenfunctions. It is done through analytic estimates of Morrey-Nirenberg and Carleman estimates.
A thin anisotropic elastic plate clamped along its lateral side and also supported at a small area $θ_{h}$ of one base is considered; the diameter of $θ_{h}$ is of the same order as the plate relative thickness $h\ll1$. In addition to the standard Kirchhoff model with the Sobolev point condition, a three-dimensional boundary layer is investigated in the vicinity of the support $θ_{h}$, which with the help of the derived weighted inequality of Korn's type, will provide an error estimate with the bound $ch^{1/2}|\ln h|$. Ignoring this boundary layer effect reduces the precision order down to $|\…
In general, for higher order elliptic equations and boundary value problems like the biharmonic equation and the linear clamped plate boundary value problem neither a maximum principle nor a comparison principle or -- equivalently -- a positivity preserving property is available. The problem is rather involved since the clamped boundary conditions prevent the boundary value problem {from} being reasonably written as a system of second order boundary value problems. It is shown that, on the other hand, for bounded smooth domains $Ω\subset\mathbb{R}^n$, the negative part of the corresponding Gre…
We consider a nonlinear (Berger or Von Karman) clamped plate model with a {\em piston-theoretic} right hand side---which include non-dissipative, non-conservative lower order terms. The model arises in aeroelasticity when a panel is immersed in a high velocity linear potential flow; in this case the effect of the flow can be captured by a dynamic pressure term written in terms of the material derivative of the plate's displacement. The effect of fully-supported internal damping is studied for both Berger and von Karman dynamics. The non-dissipative nature of the dynamics preclude the use of st…
We present a comprehensive review of the nodal domains and lines of quantum billiards, emphasizing a quantitative comparison of theoretical findings to experiments. The nodal statistics are shown to distinguish not only between regular and chaotic classical dynamics but also between different geometric shapes of the billiard system itself. We discuss, in particular, how a random superposition of plane waves can model chaotic eigenfunctions and highlight the connections of the complex morphology of the nodal lines thereof to percolation theory and Schramm-Loewner evolution. Various approaches t…
The Poisson problem consists in finding an immersed surface $Σ\subset\mathbb{R}^m$ minimising Germain's elastic energy (known as Willmore energy in geometry) with prescribed boundary, boundary Gauss map and area which constitutes a non-linear model for the equilibrium state of thin, clamped elastic plates originating from the work of S. Germain and S.D. Poisson or the early XIX century. We present a solution to this problem consisting in the minimisation of the total curvature energy $E(Σ)=\int_Σ|\operatorname{I\!I}_Σ|^2_{g_Σ}\,\mathrm{d}vol_Σ$ ($\operatorname{I\!I}_Σ$ is the second fundamenta…
