Skip to main content
SpringerLink
Log in

On the Elements Involved in the Lack of Compactness in Critical Sobolev Embedding

  • Conference paper
  • First Online:
Concentration Analysis and Applications to PDE

Part of the book series: Trends in Mathematics ((TM))

Abstract

The goal of this paper is to emphasize ideas coming from microlocal analysis to refine the study of lack of compactness in critical Sobolev embedding. In particular, we shall highlight that the elements involving in the characterization of the defect of compactness of critical Sobolev embedding in Lebesgue spaces in [13] are of different kinds than the ones intervening in the study of the critical Sobolev embedding of \( H^1(\mathbb{R}^2)\) intoth e Orlicz space \( \mathcal{L}(\mathbb{R}^2)\) in [9]; although in the two frameworks the obstruction to compactness is expressed in the same manner in terms of defect measures. This observation is based on the characterization of these embedding by means of asymptotic decompositions and puts in light the relevance of microlocal analysis behind this approach.

Mathematics Subject Classification (2010). 35A25.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. S. Adachi and K. Tanaka, Trudinger type inequalities in ℝN and their best exponents, Proceedings of the American Mathematical Society, 128 (7) (2000), 2051–2057.

    Article  MathSciNet  MATH  Google Scholar 

  2. Adimurthi and K. Tintarev, On compactness in the Trudinger–Moser inequality, arXiv:1110.3647, (2011).

    Google Scholar 

  3. H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der mathematischen Wissenschaften, Springer, (2011).

    Google Scholar 

  4. H. Bahouri, A. Cohen and G. Koch, A general wavelet-based profile decomposition in the critical embedding of function spaces, Confluentes Mathematici, 3 (3) (2011), 1–25.

    Article  MathSciNet  Google Scholar 

  5. H. Bahouri and I. Gallagher, Weak stability of the set of global solutions to the Navier-Stokes equations, arXiv:1109.4043, (2011).

    Google Scholar 

  6. H. Bahouri and P. Gérard, High frequency approximation of solutions to critical nonlinear wave equations, American Journal of Math, 121 (1999), 131–175.

    Article  MATH  Google Scholar 

  7. H. Bahouri, M. Majdoub and N. Masmoudi, On the lack of compactness in the 2D critical Sobolev embedding, Journal of Functional Analysis, 260 (2011), 208–252.

    Article  MathSciNet  MATH  Google Scholar 

  8. H. Bahouri, M. Majdoub and N. Masmoudi, Lack of compactness in the 2D critical Sobolev embedding, the general case, Notes aux Comptes-Rendus de l’Académie des Sciences de Paris, Ser. I 350, 2012, pages 177–181.

    MathSciNet  MATH  Google Scholar 

  9. H. Bahouri, M. Majdoub and N. Masmoudi, Lack of compactness in the 2D critical Sobolev embedding, the general case, arXiv:1112.2998v1.

    Google Scholar 

  10. J. Ben Ameur, Description du défaut de compacité de l’injection de Sobolev sur le groupe de Heisenberg, Bulletin de la Société Mathématique de Belgique, 15–4, 2008, pages 599–624.

    MathSciNet  Google Scholar 

  11. H. Brezis and J.M. Coron, Convergence of solutions of H-Systems or how to blow bubbles, Archiv for Rational Mechanics and Analysis, 89, 1985, pages 21–86.

    Article  MathSciNet  MATH  Google Scholar 

  12. P. Gérard, A microlocal version of concentration-compactness, Partial differential equations and mathematical physics (Copenhagen, 1995; Lund, 1995), 143–157, Progr. Nonlinear Differential Equations Appl., 21, Birkh¨auser Boston, Boston, MA, (1996).

    Google Scholar 

  13. P. Gérard, Description du défaut de compacité de l’injection de Sobolev, ESAIM: Control, Optimisation and Calculus of Variations, 3 (1998), 213–233 (electronic, URL: http://www.emath.fr/cocv/).

  14. A. Henrot and M. Pierre, Variations et optimistation de formes, Mathématiques et applications, Springer, 48, (2005).

    Google Scholar 

  15. S. Ibrahim and M. Majdoub, Comparaison des ondes linéaires et non linéaires à coefficients variables, Bulletin de la Société Mathématique de Belgique, 10 (2003), 299–312.

    MathSciNet  MATH  Google Scholar 

  16. S. Jaffard, Analysis of the lack of compactness in the critical Sobolev embeddings, Journal of Functional Analysis, 161 (1999), 384–396.

    Article  MathSciNet  MATH  Google Scholar 

  17. C.E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy critical focusing non-linear wave equation, Acta Mathematica, 201 (2008), 147–212.

    Article  MathSciNet  MATH  Google Scholar 

  18. S. Keraani, On the defect of compactness for the Strichartz estimates of the Schrödinger equation, Journal of Differential equations, 175 (2001), 353–392.

    Article  MathSciNet  MATH  Google Scholar 

  19. C. Laurent, On stabilization and control for the critical Klein–Gordon equation on a 3D compact manifold, to appear in Journal of Functional Analysis.

    Google Scholar 

  20. N.N. Lebedev, Special functions and their applications, Prentice Hall, 1965.

    Google Scholar 

  21. P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. I., Revista Matemática Iberoamericana, 1 (1985), 145–201.

    Article  MATH  Google Scholar 

  22. P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, I, Ann. Inst. H. Po incaré Anal. Non Linéaire 1 (1984), 109–145.

    Google Scholar 

  23. G. Mancini and K. Sandeep, Moser–Trudinger inequality on conformal discs, Co mmunications in Contemporary Mathematics 12 (6) (2010), 1055–1068.

    Article  MathSciNet  MATH  Google Scholar 

  24. J. Moser, A sharp form of an inequality of N. Trudinger, Indiana University Mathematics Journal, 20 (1971), 1077–1092.

    Google Scholar 

  25. F.W.J. Olver, Asymptotics and special functions, Academic Press, 1974.

    Google Scholar 

  26. M.-M. Raoan d Z.-D. Ren, Applications of Orlicz spaces, Monographs and Textbooks in Pure and Applied Mathematics, 250 (2002), Marcel Dekker Inc.

    Google Scholar 

  27. B. Ruf, A sharp Trudinger–Moser type inequality for unbounded domains in ℝ2, Journal of Functional Analysis, 219 (2005), 340–367.

    Article  MathSciNet  MATH  Google Scholar 

  28. I. Schindler and K. Tintarev, An abstract version of the concentration compactness principle, Revista Math. Complutense, 15–2, 2002, pages 417–436.

    MathSciNet  Google Scholar 

  29. S. Solimini, A note on compactness-type properties with respect to Lorentz norms of bounded subset of a Sobolev space, Annales de l’IHP analyse non linéaire, 12 (1995), 319–337.

    MathSciNet  MATH  Google Scholar 

  30. M. Struwe, A global compactness result for boundary value problems involving limiting nonlinearities, Mathematische Zeitschrift, 187, 511–517, 1984.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. LAMA, Université Paris-Est Créteil, 61, avenue du Général de Gaulle, F-94010, Créteil Cedex, France

    Hajer Bahouri

Authors
  1. Hajer Bahouri
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hajer Bahouri .

Editor information

Editors and Affiliations

  1. Centre for Applicable Mathematics Post Bag No. 6503, GKVK Post Office, Tata Institute of Fundamental Research, Bangalore, Karnataka, India

    Adimurthi

  2. Centre for Applicable Mathematics Post Bag No. 6503, GKVK Post Office, Tata Institute of Fundamental Research, Bangalore, Karnataka, India

    K. Sandeep

  3. CeReMath, Université Toulouse I, Toulouse, France

    Ian Schindler

  4. Department of Mathematics, Uppsala University, Uppsala, Sweden

    Cyril Tintarev

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer Basel AG

About this paper

Cite this paper

Bahouri, H. (2013). On the Elements Involved in the Lack of Compactness in Critical Sobolev Embedding. In: Adimurthi, ., Sandeep, K., Schindler, I., Tintarev, C. (eds) Concentration Analysis and Applications to PDE. Trends in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0373-1_1

Download citation

Publish with us

Policies and ethics

Search

Navigation