The spectrum asymptotics for the Dirichlet problem in the case of the biharmonic operator in a domain with highly indented boundary (original) (raw)
Related papers
Asymptotics for eigenvalues of the Laplacian in higher dimensional periodically perforated domains
Zeitschrift für angewandte Mathematik und Physik, 2010
This paper considers the periodic spectral problem associated with the Laplace operator written in R N (N = 3, 4, 5) periodically perforated by balls, and with homogeneous Dirichlet condition on the boundary of holes. We give an asymptotic expansion for all simple eigenvalues as the size of holes goes to zero. As an application of this result, we use Bloch waves to find the classical strange term in homogenization theory, as the size of holes goes to zero faster than the microstructure period.
Eigenvalue problem in two dimensions for an irregular boundary: Neumann condition
The European Physical Journal Plus
We formulate a systematic elegant perturbative scheme for determining the eigenvalues of the Helmholtz equation (∇ 2 + k 2 )ψ = 0 in two dimensions when the normal derivative of ψ vanishes on an irregular closed curve. The unique feature of this method, unlike other perturbation schemes, is that it does not require a separate formalism to treat degeneracies. Degenerate states are handled equally elegantly as the non-degenerate ones. A real parameter, extracted from the parameters defining the irregular boundary, serves as a perturbation parameter in this scheme as opposed to earlier schemes where the perturbation parameter is an artificial one. The efficacy of the proposed scheme is gauged by calculating the eigenvalues for elliptical and supercircular boundaries and comparing with the results obtained numerically. We also present a simple and interesting semi-empirical formula, determining the eigenspectrum of the 2D Helmholtz equation with the Dirichlet or the Neumann condition for a supercircular boundary. A comparison of the eigenspectrum for several low-lying modes obtained by employing the formula with the corresponding numerical estimates shows good agreement for a wide range of the supercircular exponent. a The European Physical Journal Plus has witnessed a flurry of activities of late is the study of quantum dots. The dots are usually taken to have a circular symmetry. But in practice that can hardly be guaranteed. There are bound to be small departures from exact circular symmetry. In such a scenario a very natural extension is to consider the confining region to be a supercircle [1] and investigate the resulting spectrum. Recently, Bera et al. have proposed a perturbative approach to the problem where the correction terms were given by a power series expansion and the method was applied for the supercircular boundaries. This then calls for a programme to solve the Helmholtz equation for a general boundary. Chakraborty et al.
On the boundary value problem of the biharmonic operator on domains with angular corners
Mathematical Methods in the …, 1980
The paper is concerned with boundary singularities of weak solutions of boundary value problems governed by the biharmonic operator. The presence of angular corner points or points at which the type of boundary condition changes in general causes local singularities in the solution. For that case the general theory of V. A. Kondrat'ev provides a priori estimates in weighted Sobolev norms and asymptotic singular representations for the solution which essentially depend on the zeros of certain transcendental functions. The distribution of these zeros will be analysed in detail for the biharmonic operator under several boundary conditions. This leads to sharp a priori estimates in weighted Sobolev norms where the weight function is characterized by the inner angle of the boundary comer. Such estimates for "negative" Sobolev norms are used to analyse also weakly nonlinear perturbations of the biharmonic operator as. for instance, the von KhnAn model in plate bending theory and the stream function formulation of the steady state Navier-Stokes problem. It turns out that here the structure of the comer singularities is essentially the same as in the corresponding linear problem.
Comptes Rendus Mécanique, 2008
We study the asymptotic behavior of the eigenelements of the Dirichlet problem for the Laplacian in a bounded domain, a part of whose boundary, depending on a small parameter ε, is highly oscillating; the frequency of oscillations of the boundary is of order ε and the amplitude is fixed. We present second-order asymptotic approximations, as ε → 0, of the eigenelements in the case of simple eigenvalues of the limit problem. To cite this article: Y. Amirat et al., C. R. Mecanique 336 (2008). Résumé Approximation asymptotique des éléments propres du problème de Dirichlet pour le Laplacien dans un domaine à frontière fortement oscillante. Nous étudions le comportement asymptotique des éléments propres du problème de Dirichlet pour le Laplacien dans un domaine borné dont une partie de la frontière, dépendant d'un petit paramètre ε, est fortement oscillante ; la fréquence des oscillations est d'ordre ε et leur amplitude est fixe. Nous présentons des approximations asymptotiques d'ordre deux des éléments propres dans le cas de valeurs propres simples du problème limite. Pour citer cet article : Y. Amirat et al., C. R. Mecanique 336 (2008).
A sufficient condition for a discrete spectrum of the Kirchhoff plate with an infinite peak
Mathematics and Mechanics of Complex Systems, 2013
Sufficient conditions for a discrete spectrum of the biharmonic equation in a two-dimensional peak-shaped domain are established. Different boundary conditions from Kirchhoff's plate theory are imposed on the boundary and the results depend both on the type of boundary conditions and the sharpness exponent of the peak.
Hadamard asymptotics for eigenvalues of the Dirichlet Laplacian
arXiv: Analysis of PDEs, 2018
This paper is dedicated to the classical Hadamard formula for asymptotics of eigenvalues of the Dirichlet-Laplacian under perturbations of the boundary. We prove that the Hadamard formula still holds for C1C^1C1-domains with C1C^1C1-perturbations. We also derive an optimal estimate for the remainder term in the C1,alphaC^{1,\alpha}C1,alpha-case. Furthermore, if the boundary is merely Lipschitz, we show that the Hadamard formula is not valid.
Zeitschrift für Analysis und ihre Anwendungen, 2000
We calculate the main asymptotic terms for eigenvalues, both simple and multiple, and eigenfunctions of the Neumann Laplacian in a three-dimensional domain Ω(h) perturbed by a small (with diameter O(h)) Lipschitz cavern ω h in a smooth boundary ∂Ω = ∂Ω(0). The case of the hole ω h inside the domain but very close to the boundary ∂Ω is under consideration as well. It is proven that the main correction term in the asymptotics of eigenvalues does not depend on the curvature of ∂Ω while terms in the asymptotics of eigenfunctions do. The influence of the shape of the cavern to the eigenvalue asymptotics relies mainly upon a certain matrix integral characteristics like the tensor of virtual masses. Asymptotically exact estimates of the remainders are derived in weighted norms.