Partial differential equations/Calculus of variations
Spectral stability results for higher-order operators under perturbations of the domain
Comptes Rendus. Mathématique, Volume 351 (2013) no. 19-20, pp. 725-730.

We analyze the spectral behavior of higher-order elliptic operators when the domain is perturbed. We provide general spectral stability results for Dirichlet and Neumann boundary conditions. Moreover, we study the bi-harmonic operator with the so-called intermediate boundary conditions. We give special attention to this last case and analyze its behavior when the boundary of the domain has some oscillatory behavior. We will show that there is a critical oscillatory behavior and that the limit problem depends on whether we are above, below or just sitting on this critical value.

Nous analysons le comportement spectral des opérateurs elliptiques dʼordre supérieur lorsque le domaine est perturbé. Nous fournissons des résultats généraux de stabilité spectrale, pour les conditions de Dirichlet et de Neumann. Par ailleurs, nous étudions lʼopérateur bi-harmonique avec les conditions aux limites dites intermédiaires. Nous accordons une attention particulière à ce dernier cas et analysons son comportement lorsque la frontière du domaine a un comportement oscillatoire. Nous allons montrer quʼil existe un comportement oscillatoire critique et que le problème à la limite dépend de ce que nous sommes au-dessus, en dessous ou précisement sur cette valeur critique.

Published online:
DOI: 10.1016/j.crma.2013.10.001
Arrieta, José M. 1; Lamberti, Pier Domenico 2

1 Departamento de Matemática Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spain
2 Dipartimento di Matematica, Università degli Studi di Padova, Via Trieste 63, 35121 Padova, Italy
     author = {Arrieta, Jos\'e M. and Lamberti, Pier Domenico},
     title = {Spectral stability results for higher-order operators under perturbations of the domain},
     journal = {Comptes Rendus. Math\'ematique},
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     year = {2013},
     doi = {10.1016/j.crma.2013.10.001},
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Arrieta, José M.; Lamberti, Pier Domenico. Spectral stability results for higher-order operators under perturbations of the domain. Comptes Rendus. Mathématique, Volume 351 (2013) no. 19-20, pp. 725-730. doi : 10.1016/j.crma.2013.10.001.

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