|
On the asymptotic behaviour of the eigenmodes for elliptic problems in domains becoming unbounded
Author(s):
Michel
Chipot;
Arnaud
Rougirel
Journal:
Trans. Amer. Math. Soc.
360
(2008),
3579-3602.
MSC (2000):
Primary 35P15, 35B40
Posted:
January 9, 2008
Retrieve article in:
PDF DVI PostScript
Abstract |
References |
Similar articles |
Additional information
Abstract:
The aim of this work is to analyze the asymptotic behaviour of the eigenmodes of some elliptic eigenvalue problems set on domains becoming unbounded in one or several directions. In particular, in the case of a linear elliptic operator in divergence form, we prove that the sequence of the  -th eigenvalues convergences to the first eigenvalue of an elliptic problems set on the section of the domain. Moreover, an optimal rate of convergence of this sequence is given.
References:
-
- 1.
- M. Chipot: Elements of Nonlinear Analysis. Birkhäuser, 2000. MR 1801735 (2002c:35001)
- 2.
- M. Chipot:
 goes to plus infinity. Birkhäuser, 2002. MR 1999898 (2005b:35001) - 3.
- M. Chipot and A. Rougirel: On the asymptotic behaviour of the solution of elliptic problems in cylindrical domains becoming unbounded. Commun. Contemp. Math. 4, 2002, no. 1, p. 15-44. MR 1890076 (2002k:35069)
- 4.
- M. Chipot and A. Rougirel: Remarks on the asymptotic behaviour of the solution to parabolic problems in domains becoming unbounded. Nonlinear Analysis 47, 2001, no. 1, p. 3-12. MR 1970625 (2004b:35141)
- 5.
- M. Chipot and A. Rougirel: On the asymptotic behaviour of the solution of parabolic problems in cylindrical domains of large size in some directions. Discrete Contin. Dyn. Syst. Ser. B 1, 2001, no. 3, p. 319-338. MR 1849821 (2002f:35036)
- 6.
- M. Chipot and A. Rougirel: Sur le comportement asymptotique de la solution de problèmes elliptiques dans des domaines cylindriques tendant vers l'infini. C. R. Acad Sci. Paris 331, 2000, no. 3, p. 435-440. MR 1792482 (2001h:35018)
- 7.
- M. Chipot and A. Rougirel: Local stability under changes of boundary conditions at a far away location. Elliptic and parabolic problems (Rolduc/Gaeta, 2001), p. 52-65, World Sci. Publishing, 2002. MR 1937527 (2003i:35119)
- 8.
- M. Chipot and Y. Xie: On the asymptotic behaviour of the
 -Laplace equation in cylinders becoming unbounded. Nonlinear partial differential equations and their applications, p. 16-27, GAKUTO Internat. Ser. Math. Sci. Appl., 20, Gakkōtosho, Tokyo, 2004. MR 2087457 - 9.
- D. Cioranescu and P. Donato: An introduction to homogenization. Oxford Lecture Series in Mathematics and its Applications # 17, 1999. MR 1765047 (2001j:35019)
- 10.
- L.C. Evans: Partial differential equations. Graduate Studies in Mathematics #19, AMS, 1998. MR 1625845 (99e:35001)
- 11.
- R. Dautray and J.L. Lions: Mathematical analysis and numerical methods for science and technology. Springer-Verlag, 1992. MR 1156075 (92k:00006)
- 12.
- D. Gilbarg and N.S. Trudinger: Elliptic Partial Differential Equations of second order. Springer-Verlag, 1983. MR 737190 (86c:35035)
- 13.
- D. Kinderlehrer and G. Stampacchia: An introduction to Variational Inequalities and their applications. Academic Press, San Francisco-London, 1980. MR 567696 (81g:49013)
- 14.
- P.A. Raviart and J.M. Thomas: Introduction à l'analyse numérique des équations aux dérivées partielles. Masson, 1983.
- 15.
- G. Stampacchia: Equations elliptiques du second ordre à coefficients discontinus. Presse de l'Université de Montréal, 1966. MR 0251373 (40:4603)
Similar Articles:
Retrieve articles in Transactions of the American Mathematical Society
with MSC
(2000):
35P15, 35B40
Retrieve articles in all Journals with MSC
(2000):
35P15, 35B40
Additional Information:
Michel
Chipot
Affiliation:
Institut für Mathematik, Universität Zürich, Winterthurerstr. 190, CH-8057 Zürich, Switzerland
Email:
m.m.chipot@math.unizh.ch
Arnaud
Rougirel
Affiliation:
Laboratoire de Mathématiques et Applications, UMR 6086, Université de Poitiers & CNRS, Téléport 2 - BP 30179, Boulevard Marie et Pierre Curie, 86 962 Futuroscope Chasseneuil Cedex, France
Email:
rougirel@math.univ-poitiers.fr
DOI:
10.1090/S0002-9947-08-04361-4
PII:
S 0002-9947(08)04361-4
Keywords:
Eigenvalue problems,
$\ell$ goes to plus infinity
Received by editor(s):
June 27, 2005
Received by editor(s) in revised form:
April 24, 2006
Posted:
January 9, 2008
Copyright of article:
Copyright
2008,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
|
Comments: webmaster@ams.org