On the stability of some isoperimetric inequalities for the fundamental tones of free plates

Buoso, Davide; Chasman, L. Mercredi; Provenzano, Luigi

doi:10.4171/jst/214

Cited by 21 publications

(32 citation statements)

References 23 publications

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“…Problem (NBS) has been discussed in [25,28,29] for σ = 1. We point out that problem (BS λ ) with σ = λ = 0 has been introduced in [12] as the natural fourth order generalization of the classical Steklov problem for the Laplacian (see also [11]). As we shall see, problem (BS λ ) shares much more analogies with the classical Steklov problem than those already presented in [12], in particular it plays a role in describing the space γ 0 (H 2 ( )) similar to that played by the Steklov problem for the Laplacian in describing γ 0 (H 1 ( )) (cf.…”

Section: Introductionmentioning

confidence: 99%

On the explicit representation of the trace space $$H^{\frac{3}{2}}$$ and of the solutions to biharmonic Dirichlet problems on Lipschitz domains via multi-parameter Steklov problems

Lamberti

Provenzano

2021

Rev Mat Complut

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We consider the problem of describing the traces of functions in $$H^2(\Omega )$$ H 2 ( Ω ) on the boundary of a Lipschitz domain $$\Omega $$ Ω of $$\mathbb R^N$$ R N , $$N\ge 2$$ N ≥ 2 . We provide a definition of those spaces, in particular of $$H^{\frac{3}{2}}(\partial \Omega )$$ H 3 2 ( ∂ Ω ) , by means of Fourier series associated with the eigenfunctions of new multi-parameter biharmonic Steklov problems which we introduce with this specific purpose. These definitions coincide with the classical ones when the domain is smooth. Our spaces allow to represent in series the solutions to the biharmonic Dirichlet problem. Moreover, a few spectral properties of the multi-parameter biharmonic Steklov problems are considered, as well as explicit examples. Our approach is similar to that developed by G. Auchmuty for the space $$H^1(\Omega )$$ H 1 ( Ω ) , based on the classical second order Steklov problem.

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Section: Introductionmentioning

confidence: 99%

On the explicit representation of the trace space $$H^{\frac{3}{2}}$$ and of the solutions to biharmonic Dirichlet problems on Lipschitz domains via multi-parameter Steklov problems

Lamberti

Provenzano

2021

Rev Mat Complut

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“…In Section 3 we examine the problem of shape differentiability of the eigenvalues. We consider problem (10) in φ(Ω) and pull it back to Ω, where φ belongs to a suitable class of diffeomorphisms. We also derive Hadamard-type formulas for the elementary symmetric functions of the eigenvalues.…”

Section: Introductionmentioning

confidence: 99%

Analyticity and Criticality Results for the Eigenvalues of the Biharmonic Operator

Buoso

2016

Springer Proceedings in Mathematics &Amp; Statistics

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Abstract:We consider the eigenvalues of the biharmonic operator subject to several homogeneous boundary conditions (Dirichlet, Neumann, Navier, Steklov). We show that simple eigenvalues and elementary symmetric functions of multiple eigenvalues are real analytic, and provide Hadamard-type formulas for the corresponding shape derivatives. After recalling the known results in shape optimization, we prove that balls are always critical domains under volume constraint.

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“…[28,Example 2.15] where problem (1.3) with τ = 0 is referred to as the Neumann problem for the biharmonic operator. Moreover, we point out that a number of recent papers devoted to the analysis of (1.2) have con rmed that problem (1.2) can be considered as the natural Neumann problem for the biharmonic operator, see [7], [8], [10], [11], [12], [13], [14], [29]. We also refer to [22] for an extensive discussion on boundary value problems for higher order elliptic operators.…”

Section: Introductionmentioning

confidence: 86%

Spectral Analysis of the Biharmonic Operator Subject to Neumann Boundary Conditions on Dumbbell Domains

Arrieta

Ferraresso

Lamberti

2017

Integr. Equ. Oper. Theory

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We consider the biharmonic operator subject to homogeneous boundary conditions of Neumann type on a planar dumbbell domain which consists of two disjoint domains connected by a thin channel. We analyse the spectral behaviour of the operator, characterizing the limit of the eigenvalues and of the eigenprojections as the thickness of the channel goes to zero. In applications to linear elasticity, the fourth order operator under consideration is related to the deformation of a free elastic plate, a part of which shrinks to a segment. In contrast to what happens with the classical second order case, it turns out that the limiting equation is here distorted by a strange factor depending on a parameter which plays the role of the Poisson coe cient of the represented plate.

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On the stability of some isoperimetric inequalities for the fundamental tones of free plates

Cited by 21 publications

References 23 publications

On the explicit representation of the trace space $$H^{\frac{3}{2}}$$ and of the solutions to biharmonic Dirichlet problems on Lipschitz domains via multi-parameter Steklov problems

On the explicit representation of the trace space $$H^{\frac{3}{2}}$$ and of the solutions to biharmonic Dirichlet problems on Lipschitz domains via multi-parameter Steklov problems

Analyticity and Criticality Results for the Eigenvalues of the Biharmonic Operator

Spectral Analysis of the Biharmonic Operator Subject to Neumann Boundary Conditions on Dumbbell Domains

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