On the Behavior of Clamped Plates under Large Compression

  title={On the Behavior of Clamped Plates under Large Compression},
  author={Pedro R. S. Antunes and Davide Buoso and Pedro J. Freitas},
  journal={SIAM J. Appl. Math.},
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… Expand
The Bilaplacian with Robin boundary conditions
We introduce Robin boundary conditions for biharmonic operators, which are a model for elastically supported plates and are closely related to the study of spaces of traces of Sobolev functions. WeExpand
The buckling eigenvalue problem in the annulus
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 theExpand
Positivity for the clamped plate equation under high tension
In this article we consider positivity issues for the clamped plate equation with high tension γ > 0. This equation is given by ∆2u − γ∆u = f under clamped boundary conditions. Here we show, thatExpand


Spectrum of the free rod under tension and compression
ABSTRACT In this paper, we study the spectrum of the one-dimensional vibrating free rod equation under tension or compression . The eigenvalues as functions of the tension/compression parameterExpand
Buckling eigenvalues for a clamped plate embedded in an elastic medium and related questions
This paper considers the dependence of the sum of the first m eigenvalues of three classical problems from linear elasticity on a physical parameter in the equation. The paper also considersExpand
On the buckling eigenvalue problem
We prove a density result which allows us to justify the application of the method of fundamental solutions to solve the buckling eigenvalue problem of a plate. We address an example of an analyticExpand
Analyticity and Criticality Results for the Eigenvalues of the Biharmonic Operator
We consider the eigenvalues of the biharmonic operator subject to several homogeneous boundary conditions (Dirichlet, Neumann, Navier, Steklov). We show that simple eigenvalues and elementaryExpand
Optimal Bilaplacian Eigenvalues
A numerical method is developed using Hadamard's shape derivative and the method of fundamental solutions as forward solver which allows one to propose numerical candidates to be minimizers for the first ten eigenvalues of both problems. Expand
Eigenvalues of poly-harmonic operators on variable domains
We consider a class of eigenvalue problems for poly-harmonic oper- ators, including Dirichlet and buckling-type eigenvalue problems. We prove an analyticity result for the dependence of the symmetricExpand
Reviving the Method of Particular Solutions
It is explained why Fox, Henrici, and Moler's formulation of this method breaks down when applied to regions that are insufficiently simple and a modification is proposed that avoids these difficulties. Expand
Generic Simplicity of the Spectrum and Stabilization for a Plate Equation
This work shows an application of a nonstandard unique continuation property for this system that also holds generically with respect to the perturbations of the domain to a result of generic stabilization for a plate system with one dissipative boundary condition. Expand
The method of fundamental solutions applied to the calculation of eigensolutions for 2D plates
In this paper we study the application of the method of fundamental solutions (MFS) to the numerical calculation of the eigenvalues and eigenfunctions for the 2D bilaplacian in simply connectedExpand
A symmetry problem in potential theory
The proof of this result is given in Section 1 ; in Section 3 we give various generalizations to elliptic differential equations other than (1). Before turning to the detailed arguments it will be ofExpand