Eigenvalues of the Laplacian with moving mixed boundary conditions: the case of disappearing Dirichlet region

Felli, Veronica; Noris, Benedetta; Ognibene, Roberto

doi:10.1007/s00526-020-01878-3

Cited by 12 publications

(8 citation statements)

References 26 publications

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“…An example is given by a decreasing family of compact sets, see e.g. [FNO,Example 3.7]. Note that this property alone is not sufficient to have the standard (i.e.…”

Section: Resultsmentioning

confidence: 99%

Perturbed eigenvalues of polyharmonic operators in domains with small holes

Felli¹,

Romani²

2022

Preprint

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We study singular perturbations of eigenvalues of the polyharmonic operator on bounded domains under removal of small interior compact sets. We consider both homogeneous Dirichlet and Navier conditions on the external boundary, while we impose homogeneous Dirichlet conditions on the boundary of the removed set. To this aim, we develop a notion of capacity which is suitable for our higher-order context, and which permits to obtain a description of the asymptotic behaviour of perturbed simple eigenvalues in terms of a capacity of the removed set, in dependence of the respective normalized eigenfunction. Then, in the particular case of a subset which is scaling to a point, we apply a blow-up analysis to detect the precise convergence rate, which turns out to depend on the order of vanishing of the eigenfunction. In this respect, an important role is played by Hardy-Rellich inequalities in order to identify the appropriate functional space containing the limiting profile. Remarkably, for the biharmonic operator this turns out to be the same, regardless of the boundary conditions prescribed on the exterior boundary.

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“…An example is given by a decreasing family of compact sets, see e.g. [FNO,Example 3.7]. Note that this property alone is not sufficient to have the standard (i.e.…”

Section: Resultsmentioning

confidence: 99%

Perturbed eigenvalues of polyharmonic operators in domains with small holes

Felli¹,

Romani²

2022

Preprint

View full text Add to dashboard Cite

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“…However, they can be obtained by arguing as in Lanza de Cristoforis [29,Appendix C] or in [49]. We note that, if (μ o , μi , ξ) is a solution of the system ( 20)-( 21), then, by integrating (20) on ∂Ω o and by the equalities…”

Section: Analytic Representation Formulas For the Solution Of The Bou...mentioning

confidence: 99%

“…As an example, we mention the celebrated works of Cioranescu and Murat [10,11] and of Marčenko and Khruslov [33] and the more recent papers of Arrieta and Lamberti [4], Arrieta, Ferraresso, and Lamberti [3], and Ferraresso and Lamberti [22]. We also mention Bonnetier, Dapogny, and Vogelius [8] concerning small perturbations in the type of boundary conditions and Felli, Noris, and Ognibene [20,21] on disappearing Neumann or Dirichlet regions in mixed eigenvalue problems.…”

Section: Introductionmentioning

confidence: 99%

Interaction of scales for a singularly perturbed degenerating nonlinear Robin problem

Musolino¹,

Mishuris²

2022

Preprint

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We study the asymptotic behavior of the solutions of a boundary value problem for the Laplace equation in a perforated domain in R n , n ≥ 3, with a (nonlinear) Robin boundary condition on the boundary of the small hole. The problem we wish to consider degenerates under three aspects: in the limit case the Robin boundary condition may degenerate into a Neumann boundary condition, the Robin datum may tend to infinity, and the size ǫ of the small hole where we consider the Robin condition collapses to 0. We study how these three singularities interact and affect the asymptotic behavior as ǫ tends to 0, and we represent the solution and its energy integral in terms of real analytic maps and known functions of the singular perturbation parameters.

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“…As an example, we mention the celebrated works of Cioranescu and Murat [39,40] and of Marčenko and Khruslov [41], as well as the more recent articles of Arrieta and Lamberti [42], Arrieta et al [43] and Ferraresso and Lamberti [44]. We also mention the work of Bonnetier et al [45] concerning small perturbations in the type of boundary conditions and of Felli et al [46,47] on disappearing Neumann or Dirichlet regions in mixed eigenvalue problems. We observe that in the present article, the boundary of the hole depends on simply through a dilation.…”

Section: Introductionmentioning

confidence: 99%

“…We also mention the work of Bonnetier et al. [ 45 ] concerning small perturbations in the type of boundary conditions and of Felli et al [ 46 , 47 ] on disappearing Neumann or Dirichlet regions in mixed eigenvalue problems.…”

Section: Introductionmentioning

confidence: 99%

Interaction of scales for a singularly perturbed degenerating nonlinear Robin problem

Musolino

Mishuris

2022

Phil. Trans. R. Soc. A.

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We study the asymptotic behaviour of solutions of a boundary value problem for the Laplace equation in a perforated domain in R n , n ≥ 3 , with a (nonlinear) Robin boundary condition on the boundary of the small hole. The problem we wish to consider degenerates in three respects: in the limit case, the Robin boundary condition may degenerate into a Neumann boundary condition, the Robin datum may tend to infinity, and the size ϵ of the small hole where we consider the Robin condition collapses to 0. We study how these three singularities interact and affect the asymptotic behaviour as ϵ tends to 0, and we represent the solution and its energy integral in terms of real analytic maps and known functions of the singular perturbation parameters. This article is part of the theme issue ‘Non-smooth variational problems and applications’.

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Eigenvalues of the Laplacian with moving mixed boundary conditions: the case of disappearing Dirichlet region

Cited by 12 publications

References 26 publications

Perturbed eigenvalues of polyharmonic operators in domains with small holes

Perturbed eigenvalues of polyharmonic operators in domains with small holes

Interaction of scales for a singularly perturbed degenerating nonlinear Robin problem

Interaction of scales for a singularly perturbed degenerating nonlinear Robin problem

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