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International Journal of Mathematics and Mathematical Sciences
Volume 2012, Article ID 108671, 16 pages
http://dx.doi.org/10.1155/2012/108671
Research Article

Strong Unique Continuation for Solutions of a -Laplacian Problem

Mathematics Department, Universidad Autónoma Metropolitana, Avenue San Rafael Atlixco No. 186, Col. Vicentina Del. Iztapalapa, 09340 México City, DF, Mexico

Received 29 June 2012; Accepted 29 September 2012

Academic Editor: Chun-Lei Tang

Copyright © 2012 Johnny Cuadro and Gabriel López. 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. T. Carleman, “Sur un problème d'unicité pour les systèmes d' equation aux dérivées partielles à deux variables indépendantes,” Arkiv för Matematik B, vol. 26, pp. 1–9, 1939. View at Google Scholar
  2. D. Jerison and C. E. Kenig, “Unique continuation and absence of positive eigenvalues for Schrödinger operators,” Annals of Mathematics, vol. 121, no. 3, pp. 463–494, 1985, With an appendix by E. M. Stein. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. E. B. Fabes, N. Garofalo, and F.-H. Lin, “A partial answer to a conjecture of B. Simon concerning unique continuation,” Journal of Functional Analysis, vol. 88, no. 1, pp. 194–210, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. D. G. de Figueiredo and J.-P. Gossez, “Strict monotonicity of eigenvalues and unique continuation,” Communications in Partial Differential Equations, vol. 17, no. 1-2, pp. 339–346, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. A. Loulit, Inégalités avec poids et problémes de continuation unique [Ph.D. thesis], Université libre de Bruxelles, 1995.
  6. I. E. Hadi and N. Tsouli, “Strong unique continuation of eigenfunctions for p-Laplacian operator,” International Journal of Mathematics and Mathematical Sciences, vol. 25, no. 3, pp. 213–216, 2001. View at Publisher · View at Google Scholar
  7. E. Acerbi and G. Mingione, “Regularity results for a class of functionals with non-standard growth,” Archive for Rational Mechanics and Analysis, vol. 156, no. 2, pp. 121–140, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. L. Diening, Theorical and numerical results for electrorheological uids [Ph.D. thesis], University of Freiburg, Germany, 2002.
  9. T. C. Halsey, “Electrorheological fluids,” Science, vol. 258, no. 5083, pp. 761–766, 1992. View at Google Scholar · View at Scopus
  10. C. Pfeiffer, C. Mavroidis, Y. Bar-Cohen, and B. Dolgin, “Electrorheological fluid based force feedback device,” in Proceedings of the 6th SPIE Telemanipulator and Telepresence Technologies, vol. 3840, pp. 88–99, Boston, Mass, USA, July 1999. View at Scopus
  11. M. Råužička, Electrorheological Fluids: Modeling and Mathematical Theory, vol. 1748 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2000. View at Publisher · View at Google Scholar
  12. W. M. Winslow, “Induced fibration of suspensions,” Journal of Applied Physics, vol. 20, no. 12, pp. 1137–1140, 1949. View at Publisher · View at Google Scholar · View at Scopus
  13. W. Orlicz, “Uber konjugierte exponentenfolgen,” Studia Mathematica, vol. 3, pp. 200–211, 1931. View at Google Scholar
  14. H. Nakano, Modulared Semi-Ordered Linear Spaces, Maruzen, Tokyo, Japan, 1950.
  15. J. Musielak, Orlicz Spaces and Modular Spaces, vol. 1034 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1983.
  16. J. Musielak and W. Orlicz, “On Modular Spaces,” Studia Mathematica, vol. 18, pp. 49–65, 1959. View at Google Scholar · View at Zentralblatt MATH
  17. X.-L. Fan and Q.-H. Zhang, “Existence of solutions for p(x)-Laplacian Dirichlet problem,” Nonlinear Analysis. Theory, Methods & Applications, vol. 52, no. 8, pp. 1843–1852, 2003. View at Publisher · View at Google Scholar
  18. X. Fan, Q. Zhang, and D. Zhao, “Eigenvalues of p(x)-Laplacian Dirichlet problem,” Journal of Mathematical Analysis and Applications, vol. 302, no. 2, pp. 306–317, 2005. View at Publisher · View at Google Scholar
  19. M. Mihăilescu, “Elliptic problems in variable exponent spaces,” Bulletin of the Australian Mathematical Society, vol. 74, no. 2, pp. 197–206, 2006. View at Publisher · View at Google Scholar
  20. O. Kováčik and J. Rákosník, “On spaces Lp(x) and W1,p(x),” Czechoslovak Mathematical Journal, vol. 41, no. 4, pp. 592–618, 1991. View at Google Scholar · View at Zentralblatt MATH
  21. D. E. Edmunds, J. Lang, and A. Nekvinda, “On Lp(x) norms,” The Royal Society of London. Proceedings. Series A, vol. 455, no. 1981, pp. 219–225, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. X. Fan, J. Shen, and D. Zhao, “Sobolev embedding theorems for spaces Wk,p(x)(Ω),” Journal of Mathematical Analysis and Applications, vol. 262, no. 2, pp. 749–760, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. L. Diening, P. Harjulehto, P. Hästö, and M. Råužička, Lebesgue and Sobolev Spaces with Variable Exponents, vol. 2017 of Lecture Notes in Mathematics, Springer, Heidelberg, Germany, 2011. View at Publisher · View at Google Scholar
  24. P. Harjulehto, P. Hästö, and V. Latvala, “Harnack's inequality for p(x)-harmonic functions with unbounded exponent p,” Journal of Mathematical Analysis and Applications, vol. 352, no. 1, pp. 345–359, 2009. View at Publisher · View at Google Scholar
  25. W. Allegretto, “Form estimates for the p(x)-Laplacean,” Proceedings of the American Mathematical Society, vol. 135, no. 7, p. 2177–2185 (electronic), 2007. View at Publisher · View at Google Scholar
  26. M. Mihăilescu, V. Rădulescu, and D. Stancu-Dumitru, “A Caffarelli-Kohn-Nirenberg-type inequality with variable exponent and applications to PDEs,” Complex Variables and Elliptic Equations, vol. 56, no. 7-9, pp. 659–669, 2011. View at Publisher · View at Google Scholar
  27. C. L. Fefferman, “The uncertainty principle,” Bulletin of the American Mathematical Society, vol. 9, no. 2, pp. 129–206, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  28. M. Schechter, Spectra of Partial Differential Operators, vol. 14, North-Holland Publishing, New York, NY, USA, 2nd edition, 1986.
  29. F. Chiarenza and M. Frasca, “A remark on a paper by C. Fefferman,” Proceedings of the American Mathematical Society, vol. 108, no. 2, pp. 407–409, 1990. View at Publisher · View at Google Scholar
  30. P. Zamboni, “Unique continuation for non-negative solutions of quasilinear elliptic equations,” Bulletin of the Australian Mathematical Society, vol. 64, no. 1, pp. 149–156, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  31. J. Cuadro and G. Lopez, “Unique Continuation of a p(x)-Lapalcian Equations,” Electronic Journal of Differential Equations, vol. 2012, no. 7, pp. 1–12, 2012. View at Google Scholar
  32. H. Brezis, Analyse Fonctionnelle: Théorie et Applications, Collection Mathématiques Appliquées pour la Maîtrise, Masson, Paris, France, 1983.
  33. O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, NY, USA, 1968.