On asymptotic properties of biharmonic Steklov eigenvalues

Liu, Genqian

doi:10.1016/j.jde.2016.07.004

Cited by 17 publications

(12 citation statements)

References 29 publications

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“…We state the following theorem, whose proof is omitted since it follows exactly the same lines as those of Theorems 1.1 and 1.2 of [29].…”

Section: Moreover the Solution U Is Uniquementioning

confidence: 99%

“…Remark B. 2 We remark that the approach used in [29] requires that the boundary of is of class C ∞ . However, as proved by another technique in [28], the asymptotic formulas for the eigenvalues of (DBS) and (NBS) when σ = 1 hold when is of class C 2 .…”

Section: Moreover the Solution U Is Uniquementioning

confidence: 99%

“…It is proved in [28,29] that if is a bounded domain in R N with C ∞ boundary, then the eigenvalues of problems (DBS) and (NBS) with σ = 1 satisfy the following asymptotic laws…”

Section: Proofmentioning

confidence: 99%

“…Problem (DBS) for σ = 1 has been studied by many authors (see e.g., [3,9,[16][17][18]25,29]); for the case σ = 1 we refer to [10], see also [4,17] for σ = 0. 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]).…”

Section: Introductionmentioning

confidence: 99%

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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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“…We state the following theorem, whose proof is omitted since it follows exactly the same lines as those of Theorems 1.1 and 1.2 of [29].…”

Section: Moreover the Solution U Is Uniquementioning

confidence: 99%

Section: Moreover the Solution U Is Uniquementioning

confidence: 99%

“…It is proved in [28,29] that if is a bounded domain in R N with C ∞ boundary, then the eigenvalues of problems (DBS) and (NBS) with σ = 1 satisfy the following asymptotic laws…”

Section: Proofmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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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“…Problem (2.11) was first introduced in [42], and has recently gained attention for its relations with the other biharmonic problems, as well as related questions (see e.g., [11,43,44]). Problem (2.11) admits as well a non-decreasing sequence of eigenvalues of finite multiplicities, diverging to plus infinity, and the corresponding eigenfunctions can be normalized to form an orthonormal basis of H 2 * (Ω).…”

Section: Functional Analytic Setupmentioning

confidence: 99%

Bulk-Boundary eigenvalues for Bilaplacian problems

Buoso¹,

Falcó²,

González³

et al. 2021

Preprint

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We initiate the study of a bulk-boundary eigenvalue problem for the Bilaplacian with a particular third order boundary condition that arises from the study of dynamical boundary conditions for the Cahn-Hilliard equation. First we consider continuity properties under parameter variation (in which the parameter also affects the domain of definition of the operator). Then we look at the ball and the annulus geometries (together with the punctured ball), obtaining the eigenvalues as solutions of a precise equation involving special functions. An interesting outcome of our analysis in the annulus case is the presence of a bifurcation from the zero eigenvalue depending on the size of the annulus.

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Spectral invariants of the perturbed polyharmonic Steklov problem

Liu

2022

Calc. Var.

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On asymptotic properties of biharmonic Steklov eigenvalues

Cited by 17 publications

References 29 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

Bulk-Boundary eigenvalues for Bilaplacian problems

Spectral invariants of the perturbed polyharmonic Steklov problem

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