A few shape optimization results for a biharmonic Steklov problem

Buoso, Davide; Provenzano, Luigi

doi:10.1016/j.jde.2015.03.013

Cited by 49 publications

(78 citation statements)

References 35 publications

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“…We refer to the book for more information on shape optimization problems for the biharmonic operator. We also refer to , where it is discussed the dependence of the eigenvalues of polyharmonic operators upon variation of the mass density, and to where the authors consider Neumann and Steklov‐type eigenvalue problems for the biharmonic operator with particular attention to shape optimization and mass concentration phenomena. We also mention , where the author considers the shape sensitivity problem for the eigenvalues of the biharmonic operator (in particular, also those of problem ) for

σ \in] - \frac{1}{N - 1}, 1 [

.…”

Section: Introductionmentioning

confidence: 99%

A note on the Neumann eigenvalues of the biharmonic operator

Provenzano

2016

Math Methods in App Sciences

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We study the dependence of the eigenvalues of the biharmonic operator subject to Neumann boundary conditions on the Poisson's ratio σ. In particular, we prove that the Neumann eigenvalues are Lipschitz continuous with respect to σ∈[0,1[and that all the Neumann eigenvalues tend to zero as σ→1−. Moreover, we show that the Neumann problem defined by setting σ = 1 admits a sequence of positive eigenvalues of finite multiplicity that are not limiting points for the Neumann eigenvalues with σ∈[0,1[as σ→1− and that coincide with the Dirichlet eigenvalues of the biharmonic operator. Copyright © 2016 John Wiley & Sons, Ltd.

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σ \in] - \frac{1}{N - 1}, 1 [

.…”

Section: Introductionmentioning

confidence: 99%

A note on the Neumann eigenvalues of the biharmonic operator

Provenzano

2016

Math Methods in App Sciences

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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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“…We note that the analysis of the cases α ≤3/2 is in spirit of the paper , which is devoted to the Navier–Stokes system. For recent results concerning domain perturbation problems for higher order operators, we refer to . We believe that our analysis could be carried out in the case of general polyharmonic operators of order 2 m subject to various types of intermediate boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

“…We note that the analysis of the cases˛Ä 3=2 is in spirit of the paper [9], which is devoted to the Navier-Stokes system. For recent results concerning domain perturbation problems for higher order operators, we refer to [4][5][6][7][8]. We believe that our analysis could be…”

Section: Introductionmentioning

confidence: 99%

Boundary homogenization for a triharmonic intermediate problem

Arrieta

Ferraresso

Lamberti

2016

Math Methods in App Sciences

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We consider the triharmonic operator subject to homogeneous boundary conditions of intermediate type on a bounded domain of the N‐dimensional Euclidean space. We study its spectral behaviour when the boundary of the domain undergoes a perturbation of oscillatory type. We identify the appropriate limit problems that depend on whether the strength of the oscillation is above or below a critical threshold. We analyse in detail the critical case that provides a typical homogenization problem leading to a strange boundary term in the limit problem. Copyright © 2016 John Wiley & Sons, Ltd.

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A few shape optimization results for a biharmonic Steklov problem

Cited by 49 publications

References 35 publications

A note on the Neumann eigenvalues of the biharmonic operator

A note on the Neumann eigenvalues of the biharmonic operator

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

Boundary homogenization for a triharmonic intermediate problem

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