Asymptotic Property of Eigenvalues and Eigenfunctions of the Laplace Operator in Domain with a Perturbed Boundary

2000 Mathematics Subject Classification: 35J05, 35C15, 44P05In this paper, we consider the variations of eigenvalues and eigenfunctions for the Laplace operator with homogeneous Dirichlet boundary conditions under deformation of the underlying domain of definition. We derive recursive formulas for the Taylor coefficients of the eigenvalues as functions of the shape-perturbation parameter and we establish the existence of a set of eigenfunctions that is jointly holomorphic in the spatial and boundary-variation variables. Using integral equations, we show that these eigenvalues are exactly built with the characteristic values of some meromorphic operator-valued functions

On the perturbation of the electromagnetic energy due to the presence of small inhomogeneities

We consider solutions to the time-harmonic Maxwell problem in $\R^3$. For such solution we provide a rigorous derivation of the asymptotic expansions in the practically interesting situation, where a finite number of inhomogeneities of small diameter are imbedded in the entire space. Then, we describe the behavior of the electromagnetic energy caused by the presence of these inhomogeneities

The integral equation methods for the perturbed Helmholtz eigenvalue problems

It is well known that the main difficulty in solving eigenvalue problems under shape deformation relates to the continuation of multiple eigenvalues of the unperturbed configuration. These eigenvalues may evolve, under shape deformation, as separated, distinct eigenvalues. In this paper, we address the integral equation method in the evaluation of eigenfunctions and the corresponding eigenvalues of the two-dimensional Laplacian operator under boundary variations of the domain. Using surface potentials, we show that the eigenvalues are the characteristic values of meromorphic operator-valued functions

Une méthode asymptotique pour les équations de Maxwell dans un milieu avec une petite hétérogénéité

In this paper we consider solutions to the perturbed Maxwell's equations in R d , for d = 2, 3. For such solutions we provide a complete asymptotic expansions of the (geometric) perturbations resulting from the presence of diametrically small heterogeneity with parameters different from the background medium. Our derivation is rigorous and is based on layer potential techniques. Our formulas extend those already derived before for the scalar cases, and may be expected to lead effective computational identification algorithms, aimed at reconstructing small dielectric object from electromagnetic boundary measurements.Dans cet article, nous considérons des solutions des équations de Maxwell perturbées dans R d, pour d = 2, 3. Pour de telles solutions, nous fournissons des développements asymptotiques complètes des perturbations (géométriques) résultant de la présence d'une hétérogénéité diamétralement petite avec des paramètres différents du milieu de référence. Notre dérivation est rigoureuse et repose sur des techniques des opérateurs de couches. Nos formules étendent celles déjà obtenues précédemment pour les cas scalaires, et on peut s'attendre à ce qu'elles conduisent des algorithmes d'identification computationnelle efficaces, visant à reconstruire un petit objet diélectrique à partir des mesures sur les bords

Diffraction d'ondes électromagnétiques par des inhomogénéités diélectriques

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