Viewing the Steklov Eigenvalues of the Laplace Operator as Critical Neumann Eigenvalues

Lamberti, Pier Domenico; Provenzano, Luigi

doi:10.1007/978-3-319-12577-0_21

Cited by 30 publications

(39 citation statements)

References 8 publications

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“…Using a mass perturbation argument, they prove that the Steklov problem can in fact be viewed as a limiting Neumann problem where the mass is distributed only on the boundary. Note that this construction was already performed in [19] for the Laplace operator, obtaining similar results (see also [16,20] for the computation of the topological derivative). Moreover, this justifies the fact of thinking of Steklov problems in terms of vibrating objects (plates or membranes) where the mass lies only on the boundary (see [23]).…”

Section: Introductionmentioning

confidence: 56%

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

2018

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We provide a quantitative version of the isoperimetric inequality for the fundamental tone of a biharmonic Neumann problem. Such an inequality has been recently established by Chasman adapting Weinberger's argument for the corresponding second order problem. Following a scheme introduced by Brasco and Pratelli for the second order case, we prove that a similar quantitative inequality holds also for the biharmonic operator. We also prove the sharpness of both such an inequality and the corresponding one for the biharmonic Steklov problem.2010 Mathematics Subject Classification. Primary 35J30. Secondary 35P15, 49R50, 74K20.

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Section: Introductionmentioning

confidence: 56%

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

2018

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“…We refer e.g., to [9] for a more detailed discussion on the Steklov eigenvalue problem for the biharmonic operator. We have the following theorem: We refer to [33,34] for the proof of Theorem 4.18 in the case of the Laplace operator and to [9] for the proof of Theorem 4.18 in the case of the biharmonic operator, and for more information on the convergence of Neumann eigenvalues to Steklov eigenvalues via mass concentration to the boundary. We remark that the proof of Theorem 4.18 for all values of m ∈ N follows exactly the same lines as the proof of the case m = 1 and m = 2.…”

Section: 3mentioning

confidence: 99%

Eigenvalues of elliptic operators with density

Colbois

Provenzano

2018

Calc. Var.

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We consider eigenvalue problems for elliptic operators of arbitrary order 2m subject to Neumann boundary conditions on bounded domains of the Euclidean N -dimensional space. We study the dependence of the eigenvalues upon variations of mass density and in particular we discuss the existence and characterization of upper and lower bounds under both the condition that the total mass is fixed and the condition that the L N 2m -norm of the density is fixed. We highlight that the interplay between the order of the operator and the space dimension plays a crucial role in the existence of eigenvalue bounds.

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“…However, instead of studying each eigenvalue problem individually, it is also interesting to explore relationships and inequalities between eigenvalues of different eigenvalue problems. Among this type of results, one can mention the relationships between the Laplace and Steklov eigenvalues studied in [21,11,18], and various inequalities between the first nonzero eigenvalue of problems (1.2)-(1.5) on bounded domains of R 2 obtained by Kuttler and Sigilito in [10]; see Table 1 (Note that there was a misprint in Inequality VI in [10]. The correct version of the inequality is stated in Table 1.).…”

Section: Introductionmentioning

confidence: 99%