Spectral stiff problems in domains surrounded by thin bands: Asymptotic and uniform estimates for eigenvalues

Gómez, D.; Лобо, М.; Назаров, С. А.; Pérez, Eugenia

doi:10.1016/j.matpur.2005.10.013

Cited by 38 publications

(47 citation statements)

References 11 publications

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“…We refer to [5,34] for similar discussions concerning second order problems. We observe that the asymptotic analysis of mass concentration problems for second order operators has been performed by several authors by exploiting asymptotic expansions methods, see e.g., [23,24] and the references therein. We also mention the alternative approach based on potential theory and functional analysis proposed in [15,35].…”

Section: Characterization Of the Spectrum Alternative Derivation Of mentioning

confidence: 99%

A few shape optimization results for a biharmonic Steklov problem

Buoso

Provenzano

2015

Journal of Differential Equations

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Abstract:We derive the equation of a free vibrating thin plate whose mass is concentrated at the boundary, namely a Steklov problem for the biharmonic operator. We provide Hadamardtype formulas for the shape derivatives of the corresponding eigenvalues and prove that balls are critical domains under volume constraint. Finally, we prove an isoperimetric inequality for the first positive eigenvalue.

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Section: Characterization Of the Spectrum Alternative Derivation Of mentioning

confidence: 99%

A few shape optimization results for a biharmonic Steklov problem

Buoso

Provenzano

2015

Journal of Differential Equations

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“…Existence and uniqueness of the second order wall-law. The boundary conditions of problem (13) contain second order normal derivatives (in the literature, these are called conditions of Wentzel type [23,6,14]). In this framework existence and uniqueness are not so standard: we provide them here for the sake of being self-contained.…”

Section: The Averaged Wall-laws: a New Derivation Processmentioning

confidence: 99%

High order multi-scale wall-laws, Part I: The periodic case

Bresch

Milisic

2010

Quart. Appl. Math.

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Abstract. In this work we present new wall-laws boundary conditions including microscopic oscillations. We consider a Newtonian flow in domains with periodic rough boundaries that we simplify considering a Laplace operator with periodic inflow and outflow boundary conditions. Following the previous approaches, see [A. Mikelić, W. Jäger, J. Diff. Eqs, 170, 96-122, (2001)] and [Y. Achdou et al, J. Comput. Phys., 147, 1, 187-218, (1998)], we construct high order boundary layer approximations and rigorously justify their rates of convergence with respect to (the roughness' thickness). We establish mathematically a poor convergence rate for averaged second order wall-laws as it was illustrated numerically for instance in [Y. Achdou, et al]. In comparison, we establish exponential error estimates in the case of an explicit multi-scale ansatz. This motivates our study to derive implicit first order multi-scale wall-laws and to show that their rate of convergence is at least of order 3 2 . We provide a numerical assessment of the claims as well as a counterexample that makes evident the impossibility of an averaged second order wall-law. Our paper may be seen as the first one to derive efficient high order wall-laws boundary conditions.

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“…They only used H 1 -estimate of eigenfunctions which is easily obtained by the variational characterization of the eigenvalues. Regarding other two-phase eigenvalue problems in this direction, we refer to [3] [4] [6].…”

Section: Figure 1: Problem Settingmentioning

confidence: 99%