Comptes Rendus
no. 3-4
Partial Differential Equations
Lack of compactness in the 2D critical Sobolev embedding, the general case
[Défaut de compacité de lʼinjection de Sobolev critique en deux dimensions dʼespace]

Présenté par : Haïm Brezis

Hajer Bahouri1 ; Mohamed Majdoub2 ; Nader Masmoudi3
1 LAMA UMR CNRS 8050, Université Paris-Est Créteil, 61, avenue du Général de Gaulle, 94010 Créteil cedex, France
2 Département de mathématiques, faculté des sciences de Tunis, 2092 Manar, Tunisia
3 The Courant Institute of Mathematical Sciences, NY University, 251 Mercer St., New York, NY 10012, USA
Comptes Rendus. Mathématique, Volume 350 (2012) no. 3-4, pp. 177-181.

Cette Note est consacrée à lʼétude du défaut de compacité de lʼinjection de Sobolev de H1(R2) dans lʼespace dʼOrlicz critique L(R2). Nous démontrons que la déscription donnée dans Bahouri et al. (2011) [5] concernant le cas radial reste valable dans le cas général (à des translations près par des coeurs de concentration). La preuve utilise des arguments de capacité ainsi quʼun processus dʼextraction de concentrations.

This Note is devoted to the description of the lack of compactness of the Sobolev embedding of H1(R2) in the critical Orlicz space L(R2). It turns out that up to cores our result is expressed in terms of the concentration-type examples derived by J. Moser (1971) in [16] as in the radial setting investigated in Bahouri et al. (2011) [5]. However, the analysis we used in this work is strikingly different from the one conducted in the radial case which is based on an L∞ estimate far away from the origin and which is no longer valid in the general frame work. The strategy we adopted to build the profile decomposition in terms of examples by Moser concentrated around cores is based on capacity arguments and relies on an extraction process of mass concentrations.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2012.01.016
Hajer Bahouri 1 ; Mohamed Majdoub 2 ; Nader Masmoudi 3

1 LAMA UMR CNRS 8050, Université Paris-Est Créteil, 61, avenue du Général de Gaulle, 94010 Créteil cedex, France
2 Département de mathématiques, faculté des sciences de Tunis, 2092 Manar, Tunisia
3 The Courant Institute of Mathematical Sciences, NY University, 251 Mercer St., New York, NY 10012, USA 
@article{CRMATH_2012__350_3-4_177_0,
     author = {Hajer Bahouri and Mohamed Majdoub and Nader Masmoudi},
     title = {Lack of compactness in the {2D} critical {Sobolev} embedding, the general case},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {177--181},
     publisher = {Elsevier},
     volume = {350},
     number = {3-4},
     year = {2012},
     doi = {10.1016/j.crma.2012.01.016},
     language = {en},
}
TY  - JOUR
AU  - Hajer Bahouri
AU  - Mohamed Majdoub
AU  - Nader Masmoudi
TI  - Lack of compactness in the 2D critical Sobolev embedding, the general case
JO  - Comptes Rendus. Mathématique
PY  - 2012
SP  - 177
EP  - 181
VL  - 350
IS  - 3-4
PB  - Elsevier
DO  - 10.1016/j.crma.2012.01.016
LA  - en
ID  - CRMATH_2012__350_3-4_177_0
ER  - 
%0 Journal Article
%A Hajer Bahouri
%A Mohamed Majdoub
%A Nader Masmoudi
%T Lack of compactness in the 2D critical Sobolev embedding, the general case
%J Comptes Rendus. Mathématique
%D 2012
%P 177-181
%V 350
%N 3-4
%I Elsevier
%R 10.1016/j.crma.2012.01.016
%G en
%F CRMATH_2012__350_3-4_177_0
Hajer Bahouri; Mohamed Majdoub; Nader Masmoudi. Lack of compactness in the 2D critical Sobolev embedding, the general case. Comptes Rendus. Mathématique, Volume 350 (2012) no. 3-4, pp. 177-181. doi : 10.1016/j.crma.2012.01.016. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.01.016/

[1] H. Bahouri; J.-Y. Chemin; R. Danchin Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, Springer, 2011

[2] H. Bahouri; A. Cohen; G. Koch A general wavelet-based profile decomposition in the critical embedding of function spaces, Confluentes Mathematici, Volume 3 (2011) no. 3, pp. 1-25

[3] H. Bahouri; I. Gallagher Weak stability of the set of global solutions to the Navier–Stokes equations, 2011 | arXiv

[4] H. Bahouri; P. Gérard High frequency approximation of solutions to critical nonlinear wave equations, American Journal of Mathematics, Volume 121 (1999), pp. 131-175

[5] H. Bahouri; M. Majdoub; N. Masmoudi On the lack of compactness in the 2D critical Sobolev embedding, Journal of Functional Analysis, Volume 260 (2011), pp. 208-252

[6] H. Bahouri; M. Majdoub; N. Masmoudi Lack of compactness in the 2D critical Sobolev embedding, the general case, 2011 | arXiv

[7] H. Brezis; S. Wainger A note on limiting cases of Sobolev embeddings and convolution inequalities, Communications in Partial Differential Equations, Volume 5 (1980), pp. 773-789

[8] P. Gérard Description du défaut de compacité de lʼinjection de Sobolev, ESAIM: Control, Optimisation and Calculus of Variations, Volume 3 (1998), pp. 213-233 http://www.emath.fr/cocv/ (electronic)

[9] A. Henrot; M. Pierre Variations et optimisation de formes, Mathématiques et Applications, vol. 48, Springer, 2005

[10] S. Ibrahim; M. Majdoub Comparaison des ondes linéaires et non linéaires à coefficients variables, Bulletin de la Société Mathématique de Belgique, Volume 10 (2003), pp. 299-312

[11] S. Jaffard Analysis of the lack of compactness in the critical Sobolev embeddings, Journal of Functional Analysis, Volume 161 (1999), pp. 384-396

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

[13] S. Keraani On the defect of compactness for the Strichartz estimates of the Shrödinger equation, Journal of Differential Equations, Volume 175 (2001), pp. 353-392

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

[15] M. Majdoub Qualitative study of the critical wave equation with a subcritical perturbation, Journal of Mathematics Analysis and Applications, Volume 301 (2005), pp. 354-365

[16] J. Moser A sharp form of an inequality of N. Trudinger, Indiana University Mathematics Journal, Volume 20 (1971), pp. 1077-1092

[17] M.-M. Rao; Z.-D. Ren Applications of Orlicz Spaces, Monographs and Textbooks in Pure and Applied Mathematics, vol. 250, Marcel Dekker Inc., 2002

[18] B. Ruf A sharp Trudinger–Moser type inequality for unbounded domains in R2, Journal of Functional Analysis, Volume 219 (2005), pp. 340-367

[19] M. Struwe Critical points of embeddings of H01,n into Orlicz spaces, Annales de lʼInstitut Henri Poincaré Analyse Non Linéaire, Volume 5 (1988), pp. 425-464

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

Structure theorems for 2D linear and nonlinear Schrödinger equations

Hajer Bahouri

C. R. Math (2015)

On the defect of compactness in Banach spaces

Sergio Solimini; Cyril Tintarev

C. R. Math (2015)