Corpus ID: 235458219

Positivity for the clamped plate equation under high tension

  title={Positivity for the clamped plate equation under high tension},
  author={Sascha Eichmann and Reiner M. Schatzle},
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, that given a positive f , i.e. upwards pushing, we find a γ0 > 0 such that for all γ ≥ γ0 the bending u is indeed positive. This γ0 only depends on the domain and the ratio of the L1 and L∞ norm of f . In contrast to a recent result by Cassani and Tarsia, our approach is valid in all dimensions. 


Optimal estimates from below for Green functions of higher order elliptic operators with variable leading coefficients
Estimates from above and below by the same positive prototype function for suitably modified Green functions in bounded smooth domains under Dirichlet boundary conditions for elliptic operators L ofExpand
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
Maximum Principle for Higher Order Operators in General Domains
We first prove De Giorgi type level estimates for functions in $W^{1,t}(\Omega)$, $\Omega\subset\mathbb{R}^N$, with $t>N\geq 2$. This augmented integrability enables us to establish a new HarnackExpand
Die Boggio-Formel für polyharmonische Dirichlet-Probleme
  • 2019
Differences between fundamental solutions of general higher order elliptic operators and of products of second order operators
We study fundamental solutions of elliptic operators of order $$2m\ge 4$$ 2 m ≥ 4 with constant coefficients in large dimensions $$n\ge 2m$$ n ≥ 2 m , where their singularities become unbounded. ForExpand
On the Behavior of Clamped Plates under Large Compression
The asymptotic behaviour of eigen values of clamped plates under large compression is determined by relating this problem to eigenvalues of the Laplacian with Robin boundary conditions, and it is seen that these depend on the value of the compression, and start developing a boundary structure as this parameter is increased. Expand
Dominance of positivity of the Green's function associated to a perturbed polyharmonic dirichlet boundary value problem by pointwise estimates
In this work we study the behaviour of the Green function for a linear higher-order elliptic problem. More precisely, we consider the Dirichlet boundary value problem in a bounded C2m,γ-smooth domainExpand
A clamped plate with a uniform weight may change sign
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 correspondingExpand
In any dimension a “clamped plate” with a uniform weight may change sign
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 ΩExpand
The First Biharmonic Steklov Eigenvalue: Positivity Preserving and Shape Optimization
We consider the Steklov problem for the linear biharmonic equation. We survey existing results for the positivity preserving property to hold. These are connected with the first Steklov eigenvalue.Expand