In any dimension a “clamped plate” with a uniform weight may change sign

@article{Grunau2014InAD,
  title={In any dimension a “clamped plate” with a uniform weight may change sign},
  author={Hans-Christoph Grunau and Guido Sweers},
  journal={Nonlinear Analysis-theory Methods \& Applications},
  year={2014},
  volume={97},
  pages={119-124}
}
  • H. Grunau, G. Sweers
  • Published 2014
  • Mathematics
  • Nonlinear Analysis-theory Methods & Applications
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. 

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References

SHOWING 1-10 OF 45 REFERENCES
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
Positivity for perturbations of polyharmonic operators with Dirichlet boundary conditions in two dimensions
Higher order elliptic partial dierential equations with Dirichlet boundary conditions in general do not satisfy a maximum principle. Polyharmonic operators on balls are an exception. Here it is shownExpand
On domains for which the clamped plate system is positivity preserving
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 affirmativeExpand
The clamped-plate equation for the limaçon
Hadamard claimed in 1907 that the clamped-plate equation is positivity preserving for domains which are bounded by a Limaçon de Pascal. We will show that this claim is false in its full generality.Expand
Positivity and Almost Positivity of Biharmonic Green’s Functions under Dirichlet Boundary Conditions
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 aExpand
When is the first eigenfunction for the clamped plate equation of fixed sign ?
It is known that the first eigenfunction of the clamped plate equation, ∆φ = λφ in Ω with φ = ∂ ∂n φ = 0 on ∂Ω, is not necessarily of fixed sign. In this article, we survey the relations betweenExpand
ON SIGN VARIATION AND THE ABSENCE OF "STRONG" ZEROS OF SOLUTIONS OF ELLIPTIC EQUATIONS
The authors prove the existence of a convex domain G with smooth boundary for which an eigenfunction corresponding to an eigenvalue of problem with operators of elliptic type is of variable sign.Expand
Polyharmonic Boundary Value Problems: Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains
Models of Higher Order.- Linear Problems.- Eigenvalue Problems.- Kernel Estimates.- Positivity and Lower Order Perturbations.- Dominance of Positivity in Linear Equations.- Semilinear Problems.-Expand
Elliptic Boundary Value Problems in Domains with Point Singularities
Introduction Part 1: Boundary value problems for ordinary differential equations on the half-axis Elliptic boundary value problems in the half-space Elliptic boundary value problems in smooth domainsExpand
Estimates for Green function and Poisson kernels of higher-order Dirichlet boundary value problems
Pointwise estimates are derived for the kernels associated to the polyharmonic Dirichlet problem on bounded smooth domains. As a consequence one obtains optimal weighted L p -L q -regularityExpand
...
1
2
3
4
5
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