The Experts below are selected from a list of 324 Experts worldwide ranked by ideXlab platform
Guido Sweers - One of the best experts on this subject based on the ideXlab platform.
-
in any dimension a Clamped Plate with a uniform weight may change sign
Nonlinear Analysis-theory Methods & Applications, 2014Co-Authors: Hans-christoph Grunau, Guido SweersAbstract:Abstract Positivity preserving properties have been conjectured for the bilaplace Dirichlet problem in many versions. In this note we show that in any dimension there exist bounded smooth domains Ω such that even the solution of Δ 2 u = 1 in Ω with the homogeneous Dirichlet boundary conditions u = u ν = 0 on ∂ Ω is sign-changing. In two dimensions this corresponds to the Kirchhoff–Love model of a Clamped Plate with a uniform weight.
-
A Clamped Plate with a uniform weight may change sign
Discrete & Continuous Dynamical Systems - S, 2014Co-Authors: Hans-christoph Grunau, Guido SweersAbstract:It is known that the Dirichlet bilaplace boundary value problem, which is used as a model for a Clamped Plate, is not sign preserving on general domains. It is also known that the corresponding first eigenfunction may change sign. In this note we will show that even a constant right hand side may result in a sign-changing solution.
-
In any dimension a “Clamped Plate” with a uniform weight may change sign
Nonlinear Analysis: Theory Methods & Applications, 2014Co-Authors: Hans-christoph Grunau, Guido SweersAbstract:Abstract Positivity preserving properties have been conjectured for the bilaplace Dirichlet problem in many versions. In this note we show that in any dimension there exist bounded smooth domains Ω such that even the solution of Δ 2 u = 1 in Ω with the homogeneous Dirichlet boundary conditions u = u ν = 0 on ∂ Ω is sign-changing. In two dimensions this corresponds to the Kirchhoff–Love model of a Clamped Plate with a uniform weight.
-
The Clamped-Plate equation for the limaçon
Annali di Matematica Pura ed Applicata (1923 -), 2005Co-Authors: Anna Dall'acqua, Guido SweersAbstract:Hadamard claimed in 1907 that the Clamped-Plate equation is positivity preserving for domains which are bounded by a Limacon de Pascal. We will show that this claim is false in its full generality. However, we will also prove that there are nonconvex limacons for which the Clamped-Plate equation has the sign-preserving property. In fact we will give an explicit bound for the parameter of the limacon where sign change may occur.
-
On domains for which the Clamped Plate system is positivity preserving
2003Co-Authors: Guido SweersAbstract:Boggio proved in 1905 that the Clamped Plate equation is positivity preserving for a disk. It is known that on many other domains such a property fails. In this paper we will show that an affirmative result holds on still a large class of domains. We also survey the available methods in obtaining domains with such property.
Hans-christoph Grunau - One of the best experts on this subject based on the ideXlab platform.
-
in any dimension a Clamped Plate with a uniform weight may change sign
Nonlinear Analysis-theory Methods & Applications, 2014Co-Authors: Hans-christoph Grunau, Guido SweersAbstract:Abstract Positivity preserving properties have been conjectured for the bilaplace Dirichlet problem in many versions. In this note we show that in any dimension there exist bounded smooth domains Ω such that even the solution of Δ 2 u = 1 in Ω with the homogeneous Dirichlet boundary conditions u = u ν = 0 on ∂ Ω is sign-changing. In two dimensions this corresponds to the Kirchhoff–Love model of a Clamped Plate with a uniform weight.
-
In any dimension a “Clamped Plate” with a uniform weight may change sign
Nonlinear Analysis: Theory Methods & Applications, 2014Co-Authors: Hans-christoph Grunau, Guido SweersAbstract:Abstract Positivity preserving properties have been conjectured for the bilaplace Dirichlet problem in many versions. In this note we show that in any dimension there exist bounded smooth domains Ω such that even the solution of Δ 2 u = 1 in Ω with the homogeneous Dirichlet boundary conditions u = u ν = 0 on ∂ Ω is sign-changing. In two dimensions this corresponds to the Kirchhoff–Love model of a Clamped Plate with a uniform weight.
-
A Clamped Plate with a uniform weight may change sign
Discrete & Continuous Dynamical Systems - S, 2014Co-Authors: Hans-christoph Grunau, Guido SweersAbstract:It is known that the Dirichlet bilaplace boundary value problem, which is used as a model for a Clamped Plate, is not sign preserving on general domains. It is also known that the corresponding first eigenfunction may change sign. In this note we will show that even a constant right hand side may result in a sign-changing solution.
-
Nonlinear Questions in Clamped Plate Models
Milan Journal of Mathematics, 2009Co-Authors: Hans-christoph GrunauAbstract:The linear Clamped Plate boundary value problem is a classical model in mechanics. The underlying differential equation is elliptic and of fourth order. The latter is a peculiar feature with respect to which this equation differs from numerous equations in physics and engineering which are of second order. Concerning the Clamped Plate boundary value problem, “linear questions” may be considered as well understood. This changes completely as soon as one poses the simplest “nonlinear question”: What can be said about positivity preserving? Does a Plate bend upwards when being pushed upwards? It is known that the answer is “no” in general. However, there are many positivity issues as e.g. “almost positivity” to be discussed.
-
The role of positive boundary data in generalized Clamped Plate equations
Zeitschrift für angewandte Mathematik und Physik, 1998Co-Authors: Hans-christoph Grunau, Guido SweersAbstract:Positivity phenomena in higher order elliptic Dirichlet problems as e.g. in the Clamped Plate equation are in general rather subtle. It depends on the domain and on the particular form of the operator whether there are comparison principles or not. Until now most papers concentrated on positivity with respect to the right-hand side, i.e. on positivity of the Green function itself. In the present paper we focus on the role of the boundary data, i.e. on positivity of certain Poisson kernels. While it is expected that the Poisson kernel of highest order behaves similarly as the Green function, it may be surprising that for Dirichlet problems of arbitrary order and in any dimension there is also a positivity result with respect to a second Poisson kernel. Furthermore a perturbation theory for this result is developed.
Harvinder Kaur - One of the best experts on this subject based on the ideXlab platform.
-
study of the effect of thermal gradient on free vibration of Clamped visco elastic rectangular Plates with linearly thickness variation in both directions
Meccanica, 2008Co-Authors: Ankit Gupta, Harvinder KaurAbstract:The effect of thermal gradient on the free vibration of Clamped visco-elastic rectangular Plate with linearly thickness variations in both the directions has been studied here. The governing differential equation has been solved using Rayleigh-Ritz technique. The frequency equation is derived for the Clamped boundary condition on all the four edges. The effect of linear temperature variation has been considered. Deflection and time period corresponding to the first two modes of vibrations of a Clamped Plate have been computed for various values of aspect ratio, thermal constants, and taper constants.
Yvan Bonnassieux - One of the best experts on this subject based on the ideXlab platform.
-
ECC - Vibration control by sliding modes for a Clamped Plate with piezoelectric actuator and sensor
2001 European Control Conference (ECC), 2001Co-Authors: Sylvaine Leleu, Hisham Abou-kandil, Yvan BonnassieuxAbstract:The global analytical modeling of a Clamped Plate equipped with piezoelectric sensor and actuator is developed. The vibrations of the Plate are damped with a Linear Quadratic (LQ) control which exceeds the maximum voltage delivered by the piezoelectric actuator supplier, leading to a saturated imput. To deal with this problem of saturation, a sliding mode control is designed and the stability of the controlled system is examined via a Lyapunov approach.
Lingzhong Zeng - One of the best experts on this subject based on the ideXlab platform.
-
Eigenvalues for the Clamped Plate Problem of $\mathfrak{L}^{2}_{\nu}$ Operator on Complete Riemannian manifolds
arXiv: Differential Geometry, 2021Co-Authors: Lingzhong ZengAbstract:$\mathfrak{L}_{\nu}$ operator is an important extrinsic differential operator of divergence type and has profound geometric settings. In this paper, we consider the Clamped Plate problem of $\mathfrak{L}^{2}_{\nu}$ operator on a bounded domain of the complete Riemannian manifolds. A general formula of eigenvalues of $\mathfrak{L}^{2}_{\nu}$ operator is established. Applying this general formula, we obtain some estimates for the eigenvalues with higer order on the complete Riemannian manifolds. As several fascinating applications, we discuss this eigenvalue problem on the complete translating solitons, minimal submanifolds on the Euclidean space, submanifolds on the unit sphere and projective spaces. In particular, we get a universal inequality with respect to the $\mathcal{L}_{II}$ operator on the translating solitons. Usually, it is very difficult to get universal inequalities for weighted Laplacian and even Laplacian on the complete Riemannian manifolds. Therefore, this work can be viewed as a new contribution to universal inequality.
-
Eigenvalue Inequalities for the Clamped Plate Problem of $\mathfrak{L}^{2}_{\nu}$ Operator
arXiv: Differential Geometry, 2021Co-Authors: Lingzhong ZengAbstract:$\mathfrak{L}_{II}$ 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 $\mathfrak{L}_{\nu}$, and investigate the Clamped Plate problem of bi-$\mathfrak{L}_{\nu}$ operator, which is denoted by $\mathfrak{L}_{\nu}^{2}$, on the complete Riemannian manifolds. A general formula of eigenvalues for the $\mathfrak{L}_{\nu}^{2}$ operator is established. Applying this formula, we estimate the eigenvalues with lower order on the Riemannian manifolds. As some further applications, we establish some eigenvalue inequalities for this operator on the translating solitons with respect to the mean curvature flows, submanifolds of the Euclidean spaces, unit spheres and projective spaces. In particular, for the case of translating solitons, all of the eigenvalue inequalities are universal.