About this Journal Submit a Manuscript Table of Contents
International Journal of Differential Equations
Volume 2013 (2013), Article ID 210270, 4 pages
http://dx.doi.org/10.1155/2013/210270
Research Article

Behavior of the -Laplacian on Thin Domains

Universidade Estadual Paulista, Departamento de Matemática, Instituto de Geociências e Ciências Exatas, 13506-900 Rio Claro, SP, Brazil

Received 23 July 2013; Revised 2 October 2013; Accepted 3 October 2013

Academic Editor: Tuncay Candan

Copyright © 2013 Ricardo P. Silva. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. J. M. Arrieta, A. N. Carvalho, M. C. Pereira, and R. P. Silva, “Semilinear parabolic problems in thin domains with a highly oscillatory boundary,” Nonlinear Analysis, Theory, Methods and Applications, vol. 74, no. 15, pp. 5111–5132, 2011. View at Publisher · View at Google Scholar · View at Scopus
  2. J. M. Arrieta and M. C. Pereira, “Elliptic problems in thin domains with highly oscillating boundaries,” Boletín de la Sociedad EspañoLa de Matematica Aplicada, vol. 51, pp. 17–25, 2010. View at Google Scholar
  3. J. M. Arrieta and M. C. Pereira, “Homogenization in a thin domain with an oscillatory boundary,” Journal des Mathematiques Pures et Appliquees, vol. 96, no. 1, pp. 29–57, 2011. View at Publisher · View at Google Scholar · View at Scopus
  4. J. M. Arrieta and M. C. Pereira, “Thin domain with extremely high oscillatory boundaries,” Journal of Mathematical Analysis and Applications, vol. 404, no. 1, pp. 86–104, 2013. View at Google Scholar
  5. D. Cioranescu and J. S. Jean-Paulin, Homogenization of Reticulated Structures, Springer, New York, NY, USA, 1980.
  6. M. C. Pereira and R. P. Silva, “Rates of convergence for a homogenization problem in highly oscillating thin domains,” Proceeding of Dynamic Systems and Applications, vol. 6, pp. 337–340, 2012. View at Google Scholar
  7. M. C. Pereira and R. P. Silva, “Error estimatives for a Neumann problem in highly oscillating thin domain,” Discrete and Continuous Dynamical Systems A, vol. 33, no. 2, pp. 803–817, 2013. View at Google Scholar
  8. F. Antoci and M. Prizzi, “Reaction-diffusion equations on unbounded thin domains,” Topological Methods in Nonlinear Analysis, vol. 18, pp. 283–302, 2001. View at Google Scholar
  9. T. Elsken, “Limiting behavior of attractors for systems on thin domains,” Hiroshima Mathematical Journal, vol. 32, no. 3, pp. 389–415, 2002. View at Google Scholar
  10. T. Elsken, “A reaction-diffusion equation on a net-shaped thin domain,” Studia Mathematica, vol. 165, no. 2, pp. 159–199, 2004. View at Google Scholar · View at Scopus
  11. T. Elsken, “Continuity of attractors for net-shaped thin domain,” Topological Methods in Nonlinear Analysis, vol. 26, pp. 315–354, 2005. View at Google Scholar
  12. J. K. Hale and G. Raugel, “Reaction-diffusion equations on thin domains,” Journal de Mathematiques Pures et Apliquees, vol. 9, no. 71, pp. 33–95, 1992. View at Google Scholar
  13. M. Prizzi and K. P. Rybakowski, “The effect of domain squeezing upon the dynamics of reaction-diffusion equations,” Journal of Differential Equations, vol. 173, no. 2, pp. 271–320, 2001. View at Publisher · View at Google Scholar · View at Scopus
  14. G. Raugel, “Dynamics of partial differential equations on thin domains,” in Dynamical Systems, vol. 1609 of Lecture Notes in Mathematics, pp. 208–315, Springer, New York, NY, USA, 1995. View at Google Scholar
  15. R. P. Silva, “A note on resolvent convergence on a thin domain,” Bulletin of the Australian Mathematical Society, 2013. View at Publisher · View at Google Scholar
  16. V. L. Carbone, C. B. Gentile, and K. Schiabel-Silva, “Asymptotic properties in parabolic problems dominated by a p-Laplacian operator with localized large diffusion,” Nonlinear Analysis, Theory, Methods and Applications, vol. 74, no. 12, pp. 4002–4011, 2011. View at Publisher · View at Google Scholar · View at Scopus
  17. J. L. Lions, Quelques Methodes de Resolution des Problemes aux Limites non Lineaires, Dunod, Paris, France, 1969.