Comptes Rendus
Partial Differential Equations
Γ-convergence and Sobolev norms
Comptes Rendus. Mathématique, Volume 345 (2007) no. 12, pp. 679-684.

We study a Γ-convergence problem related to a new characterization of Sobolev spaces W1,p(RN) (p>1) established in H.-M. Nguyen [H.-M. Nguyen, Some new characterizations of Sobolev spaces, J. Funct. Anal. 237 (2006) 689–720] and J. Bourgain and H.-M. Nguyen [J. Bourgain, H.-M. Nguyen, A new characterization of Sobolev spaces, C. R. Acad. Sci. Paris, Ser. I 343 (2006) 75–80]. We can also handle the case p=1 which was out of reach previously.

On étudie un problème de Γ-convergence qui apparaît naturellement en liaison avec les travaux de H.-M. Nguyen [H.-M. Nguyen, Some new characterizations of Sobolev spaces, J. Funct. Anal. 237 (2006) 689–720], et J. Bourgain et H.-M. Nguyen [J. Bourgain, H.-M. Nguyen, A new characterization of Sobolev spaces, C. R. Acad. Sci. Paris, Ser. I 343 (2006), 75–80] concernant des nouvelles caractérisations des espaces de Sobolev W1,p(RN) (p>1). On peut aussi traiter le cas p=1 qui était inaccessible précédemment.

Accepted:
Published online:
DOI: 10.1016/j.crma.2007.11.005

Hoai-Minh Nguyen 1

1 Rutgers University, Department of Mathematics, Hill Center, Busch Campus, 110, Frelinghuysen Road, Piscataway, NJ 08854, USA
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Hoai-Minh Nguyen. Γ-convergence and Sobolev norms. Comptes Rendus. Mathématique, Volume 345 (2007) no. 12, pp. 679-684. doi : 10.1016/j.crma.2007.11.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.11.005/

[1] J. Bourgain; H. Brezis; P. Mironescu Another look at Sobolev spaces (J.L. Menaldi; E. Rofman; A. Sulem, eds.), Optimal Control and Partial Differential Equations, A volume in honour of A. Bensoussan's 60th birthday, 2001, pp. 439-455

[2] J. Bourgain; H. Brezis; H.-M. Nguyen A new estimate for the topological degree, C. R. Acad. Sci. Paris, Ser. I, Volume 340 (2005), pp. 787-791

[3] J. Bourgain; H.-M. Nguyen A new characterization of Sobolev spaces, C. R. Acad. Sci. Paris, Ser. I, Volume 343 (2006), pp. 75-80

[4] A. Braides Γ-Convergence for Beginners, Oxford Lecture Series in Mathematics and its Applications, vol. 22, Oxford University Press, Oxford, 2002

[5] H. Brezis How to recognize constant functions. A connections with Sobolev spaces, Volume in honor of M. Vishik, Uspekhi Mat. Nauk, Volume 57 (2002), pp. 59-74 (English translation in Russian Math. Surveys, 57, 2002, pp. 693-708)

[6] H. Brezis New questions related to the topological degree, The Unity of Mathematics, A volume in honor of ninetieth birthday of I.M. Gelfand, Progr. Math., vol. 244, Birkhäuser Boston, Boston, MA, 2006, pp. 137-154

[7] D. Chiron On the definition of Sobolev and BV spaces into metric spaces and the trace problem, Commun. Contemp. Math., Volume 7 (2007), pp. 473-513

[8] L.C. Evans; R.F. Gariepy Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton FL, 1992

[9] G. Dal Maso An Introduction to Γ-Convergence, Progress in Nonlinear Differential Equations and their Applications, vol. 8, Birkhäuser Boston, Inc., Boston, MA, 1993

[10] H.-M. Nguyen Some new characterizations of Sobolev spaces, J. Funct. Anal., Volume 237 (2006), pp. 689-720

[11] H.-M. Nguyen Optimal constant in a new estimate for the degree, J. Analyse Math., Volume 101 (2007), pp. 367-395

[12] H.-M. Nguyen Further characterizations of Sobolev spaces, J. European Math. Soc., Volume 10 (2008), pp. 191-229

[13] H.-M. Nguyen, Γ-convergence, Sobolev norms, and BV functions, in preparation

[14] H.-M. Nguyen, Some inequalities related to Sobolev norms, in preparation

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