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.

Received:
Accepted:
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
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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. http://www.numdam.org/articles/10.1016/j.crma.2013.10.001/

[1] Arrieta, J.M. Neumann eigenvalue problems on exterior perturbations of the domain, J. Differ. Equ., Volume 118 (1995), pp. 54-103

[2] Arrieta, J.M.; Carvalho, A.N. Spectral convergence and nonlinear dynamics of reaction–diffusion equations under perturbations of the domain, J. Differ. Equ., Volume 199 (2004), pp. 143-178

[3] Arrieta, J.M.; Carvalho, A.N.; Lozada-Cruz, G. Dynamics in dumbbell domains I. Continuity of the set of equilibria, J. Differ. Equ., Volume 231 (2006), pp. 551-597

[4] J.M. Arrieta, P.D. Lamberti, Higher-order operators and domain perturbation, in preparation.

[5] Babuška, I. The theory of small changes in the domain of existence in the theory of Partial Differential Equations and its applications, Prague, 1962, Publ. House Czechoslovak Acad. Sci./Academic Press, Prague/New York (1963), pp. 13-26

[6] Bucur, D.; Buttazzo, G. Variational Methods in Shape Optimization Problems, Prog. Nonlinear Differ. Equ. Appl., vol. 65, Birkhäuser, Boston, 2005

[7] Buoso, D.; Lamberti, P.D. Eigenvalues of polyharmonic operators on variable domains, ESAIM Control Optim. Calc. Var. (2013) (in press) | DOI

[8] Buoso, D.; Lamberti, P.D. Shape deformation for vibrating hinged plates, Math. Methods Appl. Sci. (2013) (in press) | DOI

[9] Burenkov, V.; Lamberti, P.D. Spectral stability of higher order uniformly elliptic operators, Sobolev Spaces in Mathematics. II, Int. Math. Ser. (N. Y.), vol. 9, Springer, New York, 2009, pp. 69-102

[10] Burenkov, V.; Lamberti, P.D. Sharp spectral stability estimates via the Lebesgue measure of domains for higher order elliptic operators, Rev. Mat. Complut., Volume 25 (2012), pp. 435-457

[11] Carvalho, A.N.; Piskarev, S. A general approximation scheme for attractors of abstract parabolic problems, Numer. Funct. Anal. Optim., Volume 27 (2006), pp. 785-829

[12] Casado-Díaz, J.; Luna-Laynez, M.; Suárez-Grau, F.J. Asymptotic behavior of the Navier–Stokes system in a thin domain with Navier condition on a slightly rough boundary, SIAM J. Math. Anal., Volume 45 (2013), pp. 1641-1674

[13] Gazzola, F.; Grunau, H.-C.; Sweers, G. Polyharmonic Boundary Value Problems. Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains, Lect. Notes Math., vol. 1991, Springer-Verlag, Berlin, 2010

[14] Mazʼya, V.G.; Nazarov, S.A. Paradoxes of limit passage in solutions of boundary value problems involving the approximation of smooth domains by polygonal domains, Math. USSR, Izv., Volume 29 (1987) no. 3, pp. 511-533

[15] Mazʼya, V.G.; Nazarov, S.A.; Plamenevskii, B. Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains I and II, Birkhäuser, Basel, 2000

[16] Stummel, F. Perturbation of domains in elliptic boundary-value problems, Lecture Notes Math., vol. 503, Springer-Verlag, Berlin–Heidelberg–New York, 1976, pp. 110-136

[17] Vainikko, G. Über die Konvergenz und Divergenz von Näherungsmethoden bei Eigenwertproblemen, Math. Nachr., Volume 78 (1977), pp. 145-164

[18] Vainikko, G. Regular convergence of operators and the approximate solution of equations, Mathematical Analysis, vol. 16, VINITI, Moscow, 1979, pp. 5-53 151 (in Russian)

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