Journal of the Korean Society for Industrial and Applied Mathematics

Korean Society for Industrial and Applied Mathematics (한국산업응용수학회)
권호 표지
DOI QR코드

DOI QR Code

ℓ GOES TO PLUS INFINITY : AN UPDATE

  • Received : 2014.04.21
  • Accepted : 2014.05.17
  • Published : 2014.06.25
Download PDF
⟨ Previous Next ⟩

Abstract

The goal of this note is to describe the asymptotic behaviour of problems set in cylinders when the size of them is becoming infinite. This leads to consider problems in unbounded domains as well as new singular perturbations issues.

Keywords

References

  1. S. Baillet, A. Henrot, T. Takahashi, Convergence results for a semilinear problem and for a Stokes problem in a periodic geometry. Asymptotic Anal. 50, (2006), 325-337.
  2. C. Baiocchi, Su un problema di frontiera libera connesso a questioni di idraulica. Ann. Mat. Pura Appl. (4) 92 (1972), 107-127. https://doi.org/10.1007/BF02417940
  3. H. Brezis, D. Kinderlehrer and G. Stampacchia, Sur une nouvelle formulation du probleme de l'ecoulement a travers une digue. C. R. Acad. Sci. Paris Sr. A-B 287 (1978), no. 9, 711-714.
  4. B. Brighi, S. Guesmia, On elliptic boundary value problems of order 2m in cylindrical domain of large size. Adv. Math. Sci. Appl. 18(1), 2008, 237-250.
  5. B. Brighi, S. Guesmia, Asymptotic behavior of solutions of hyperbolic problems on a cylindrical domain. Disc. Cont. Dyn. Syst. Suppl. 2007, 160-169.
  6. M. Chipot, Elements of Nonlinear Analysis. Birkhauser Advanced Text, 2000.
  7. M. Chipot, goes to plus infinity. Birkhauser Advanced Text, 2002.
  8. M. Chipot, Elliptic Equations: An Introductory Course. Birkhauser Advanced Text, 2009.
  9. M. Chipot, On some anisotropic singular perturbation problems. Asymptotic Analysis, 55 (2007), 125-144.
  10. M. Chipot, Some Remarks on Anisotropic Singular Perturbation Problems. Proceedings of IUTAM Symposium on the relations of shell, plate, beam and 3D models, Tbilisi, Georgia, April 2007, Springer Edt., (2008), 91-100.
  11. M. Chipot, S. Guesmia, On the asymptotic behavior of elliptic, anisotropic singular perturbations problems. Comm. Pure Applied Anal., 8, 1, (2009), 179-194.
  12. M. Chipot, S. Guesmia, Corrector for some asymptotic problems. Proceedings of the Steklov Institue of Mathematics, Vol 270, (2010), 263-277. https://doi.org/10.1134/S0081543810030211
  13. M. Chipot, S. Guesmia, On some anisotropic, nonlocal, parabolic singular perturbations problems. Applicable Analysis, Vol. 90, No. 12, (2011), 1775-1789. https://doi.org/10.1080/00036811003627542
  14. M. Chipot, S. Guesmia, On a class of integro-differential problems. Comm. Pure Applied Analysis, 9, 5, (2010), 1249-1262. https://doi.org/10.3934/cpaa.2010.9.1249
  15. M. Chipot, S. Guesmia and A. Sengouga, Singular Perturbations of Some Nonlinear Problems. Journal of Mathematical Sciences, Vol 176, 6, (2011), 828-843. https://doi.org/10.1007/s10958-011-0439-y
  16. M. Chipot, S. Mardare, Asymptotic behaviour of the Stokes problem in cylinders becoming unbounded in one direction. J. Math. Pures Appl. 90, (2008), 133-159. https://doi.org/10.1016/j.matpur.2008.04.002
  17. M. Chipot, S. Mardare, On correctors for the Stokes problem in cylinders. Proceedings of the conference on nonlinear phenomena with energy dissipation, Chiba, November 2007
  18. P. Colli and all Edts, Gakuto International Series, Mathematical Sciences and Applications, Vol 29, Gakkotosho, (2008), 37-52.
  19. M. Chipot, A. Rougirel, On the asymptotic behaviour of the solution of elliptic problems in cylindrical domains becoming unbounded. Communications in Contemporary Mathematics, Vol 4, 1, (2002), 15-44. https://doi.org/10.1142/S0219199702000555
  20. M. Chipot, A. Rougirel, On the asymptotic behaviour of the solution of parabolic problems in cylindrical domains of large size in some directions. Discrete an Continuous Dynamical Systems, Series B, 1, (2001), 319-338. https://doi.org/10.3934/dcdsb.2001.1.319
  21. M. Chipot, A. Rougirel, Remarks on the asymptotic behaviour of the solution to parabolic problems in domains becoming unbounded. Nonlinear Analysis, Vol 47, (2001), 3-11. https://doi.org/10.1016/S0362-546X(01)00151-1
  22. M. Chipot, A. Rougirel, Local stability under changes of boundary conditions at a far away location. Proceedings of the Fourth European Conference on Elliptic and Parabolic Problems, Rolduc-Gaeta 2001, World Scientific, (2002), 52-65.
  23. M. Chipot, A. Rougirel, On the asymptotic behaviour of the eigenmodes for elliptic problems in domains becoming unbounded. Trans. Amer. Math. Soc. 360 (2008), 3579-3602. https://doi.org/10.1090/S0002-9947-08-04361-4
  24. M. Chipot, P. Roy and I. Shafrir, Asymptotics of eigenstates of elliptic problems with mixed boundary data on domains tending to infinity. Asymptotic Analysis 85, (2013), 199-227.
  25. M. Chipot, Y. Xie, On the asymptotic behaviour of the p-Laplace equation in cylinders becoming unbounded. Proceedings of International Conference: Nonlinear PDE's and their Applications, N. Kenmochi, M. Otani, S. Zheng Edts., Gakkotosho, (2004), 16-27.
  26. M. Chipot, Y. Xie, On the asymptotic behaviour of elliptic problems with periodic data. C. R. Acad. Sci. Paris, Ser. I 339, (2004), 477-482. https://doi.org/10.1016/j.crma.2004.09.007
  27. M. Chipot, Y. Xie, Elliptic problems with periodic data: an asymptotic analysis. Journ.Math. Pures et Appl., 85 (2006), 345-370. https://doi.org/10.1016/j.matpur.2005.07.002
  28. M. Chipot, Y. Xie, Asymptotic behaviour of nonlinear parabolic problems with periodic data. Progress in PDE and their applications, Vol 63, Nonlinear elliptic and parabolic problems: A special tribute to the work of H. Brezis, (2005), 147-156, Birkhauser.
  29. M. Chipot, Y. Xie, Some issues on the p-Laplace equation in cylindrical domains. Proceedings of the Steklov Institue of Mathematics, 261, (2008), 287-294. https://doi.org/10.1134/S0081543808020235
  30. M. Chipot, K. Yeressian, Exponential rates of convergence by an iteration technique. C. R. Acad. Sci. Paris, Ser. I 346, (2008), 21-26. https://doi.org/10.1016/j.crma.2007.12.004
  31. M. Chipot, K. Yeressian, On the asymptotic behavior of variational inequalities set in cylinders. DCDS, Series A, Vol 33, 11 & 12, (2013), 4875-4890. https://doi.org/10.3934/dcds.2013.33.4875
  32. M. Chipot, K. Yeressian, On Some Variational Inequalities in Unbounded Domains. Boll. del UMI, (9), V, (2012), 243-262.
  33. M. Chipot, K. Yeressian, Asymptotic behaviour of the solution to variational inequalities with joint constraints on its value and its gradient. Contemporary Mathematics 594, (2013), 137-154. https://doi.org/10.1090/conm/594/11797
  34. L.C. Evans, Partial differential equations. Graduate Studies in Mathematics 19, AMS, 1998.
  35. J.-L. Lions, Quelques methodes de resolution des problemes aux limites non lineaires. Dunod-Gauthier-Villars, Paris, 1969.
  36. S. Guesmia, Some results on the asymptotic behavior for hyperbolic problems in cylindrical domains becoming unbounded. J. Math. Anal. Appl. 341(2), 2008, 1190-1212. https://doi.org/10.1016/j.jmaa.2007.11.001
  37. S. Guesmia, Asymptotic behavior of elliptic boundary-value problems with some small coefficients. Electron. J. Diff. Equ. 2008 (59), (2008), 1-13.
  38. S. Guesmia, Some convergence results for quasilinear parabolic boundary value problems in cylindrical domain of large size. Nonlinear Anal. 70(9), 2009, 3320-3331.
  39. S. Guesmia, Resultats sur l'analyse asymptotique des equations aux derivees partielles. Habilitation, Universite de Haute Alsace, France (2013).
  40. S. Guesmia, Large time and space size behaviour of the heat equation in non-cylindrical domains. Arch. Math., (2013), 101 (3), 293-299. https://doi.org/10.1007/s00013-013-0555-7
  41. S. Guesmia, A. Sengouga, Some singular perturbations results for semilinear hyperbolic problems. Discrete. Cont. Dyn. Syst. Ser. S, 5(3), 2012, 567-580.
  42. S. Guesmia, A. Sengouga, Anisotropic singular perturbations for hyperbolic problems. Appl. Math. Comput. 217 (22), (2011), 8983-8996. https://doi.org/10.1016/j.amc.2011.03.104
  43. D. Mugnai, A limit problem for degenerate quasilinear variational inequalities in cylinders. Recent trends in nonlinear partial differential equations. I. Evolution problems, Contemp. Math., 594, Amer. Math. Soc., Providence, RI, (2013), 281-293. https://doi.org/10.1090/conm/594/11795
  44. O. A. Oleinik and G. A. Yosifian, Boundary value problems for second order elliptic equations in unbounded domains and Saint-Venants principle. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 4, no. 2, (1977), 269-290.
  45. P. Roy, Some results in asymptotic analysis and nonlocal problems. Thesis University of Zurich, (2013).
  46. Y. Xie, On asymptotic problems in Cylinders and other mathematical issues. Thesis University of Zurich, (2006).
  47. K. Yeressian, Asymptotic Behavior of Elliptic Nonlocal Equations Set in Cylinders. To appear in Asymptotic Analysis.
  48. K. Yeressian, Spatial asymptotic behaviour of elliptic equations and variational inequalities. Thesis University of Zurich, (2010).

Cited by

  1. On The Asymptotic Behaviour of Some Problems of the Calculus of Variations vol.1, pp.2, 2014, https://doi.org/10.1007/bf03377383