Comptes Rendus
no. 8
Mathematical analysis/Functional analysis
On the shape factor of interaction laws for a non-local approximation of the Sobolev norm and the total variation
[Sur le facteur de forme des lois d'interaction pour une approximation non locale de la norme de Sobolev et de la variation totale]

Présenté par : Haïm Brézis

Clara Antonucci1 ; Massimo Gobbino2 ; Matteo Migliorini1 ; Nicola Picenni1
1 Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56126 Pisa, Italy
2 Dipartimento di Ingegneria Civile e Industriale, Largo Lucio Lazzarino, 56122 Pisa, Italy
Comptes Rendus. Mathématique, Volume 356 (2018) no. 8, pp. 859-864.

Nous considérons la famille des fonctionnelles non locales et non convexes introduites par H. Brézis et H.-M. Nguyen dans un article récent. Ces fonctionelles Gamma-convergent vers un multiple de la norme de Sobolev ou de la variation totale, en fonction d'un exposant de sommabilité, mais les valeurs exactes des constantes sont inconnues dans de nombreux cas.

Nous décrivons une nouvelle approche pour le résultat de Gamma-convergence, qui conduit, dans certains cas particuliers, à la valeur exacte des constantes et à l'existence de familles optimales régulières.

We consider the family of non-local and non-convex functionals introduced by H. Brézis and H.-M. Nguyen in a recent paper. These functionals Gamma-converge to a multiple of the Sobolev norm or the total variation, depending on a summability exponent, but the exact values of the constants are unknown in many cases.

We describe a new approach to the Gamma-convergence result that leads in some special cases to the exact value of the constants, and to the existence of smooth recovery families.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2018.05.014
Clara Antonucci 1 ; Massimo Gobbino 2 ; Matteo Migliorini 1 ; Nicola Picenni 1

1 Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56126 Pisa, Italy
2 Dipartimento di Ingegneria Civile e Industriale, Largo Lucio Lazzarino, 56122 Pisa, Italy 
@article{CRMATH_2018__356_8_859_0,
     author = {Clara Antonucci and Massimo Gobbino and Matteo Migliorini and Nicola Picenni},
     title = {On the shape factor of interaction laws for a non-local approximation of the {Sobolev} norm and the total variation},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {859--864},
     publisher = {Elsevier},
     volume = {356},
     number = {8},
     year = {2018},
     doi = {10.1016/j.crma.2018.05.014},
     language = {en},
}
TY  - JOUR
AU  - Clara Antonucci
AU  - Massimo Gobbino
AU  - Matteo Migliorini
AU  - Nicola Picenni
TI  - On the shape factor of interaction laws for a non-local approximation of the Sobolev norm and the total variation
JO  - Comptes Rendus. Mathématique
PY  - 2018
SP  - 859
EP  - 864
VL  - 356
IS  - 8
PB  - Elsevier
DO  - 10.1016/j.crma.2018.05.014
LA  - en
ID  - CRMATH_2018__356_8_859_0
ER  - 
%0 Journal Article
%A Clara Antonucci
%A Massimo Gobbino
%A Matteo Migliorini
%A Nicola Picenni
%T On the shape factor of interaction laws for a non-local approximation of the Sobolev norm and the total variation
%J Comptes Rendus. Mathématique
%D 2018
%P 859-864
%V 356
%N 8
%I Elsevier
%R 10.1016/j.crma.2018.05.014
%G en
%F CRMATH_2018__356_8_859_0
Clara Antonucci; Massimo Gobbino; Matteo Migliorini; Nicola Picenni. On the shape factor of interaction laws for a non-local approximation of the Sobolev norm and the total variation. Comptes Rendus. Mathématique, Volume 356 (2018) no. 8, pp. 859-864. doi : 10.1016/j.crma.2018.05.014. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.05.014/

[1] C. Antonucci; M. Gobbino; M. Migliorini; N. Picenni Optimal constants for a non-local approximation of Sobolev norms and total variation | arXiv

[2] C. Antonucci; M. Gobbino; N. Picenni On the gap between gamma-limit and pointwise limit for a non-local approximation of the total variation | arXiv

[3] J. Bourgain; H.-M. Nguyen A new characterization of Sobolev spaces, C. R. Acad. Sci. Paris, Ser. I, Volume 343 (2006) no. 2, pp. 75-80

[4] H. Brézis New approximations of the total variation and filters in imaging, Atti Accad. Naz. Lincei, Rend. Lincei, Mat. Appl., Volume 26 (2015) no. 2, pp. 223-240

[5] H. Brézis; H.-M. Nguyen Non-convex, non-local functionals converging to the total variation, C. R. Acad. Sci. Paris, Ser. I, Volume 355 (2017) no. 1, pp. 24-27

[6] H. Brézis; H.-M. Nguyen Non-local functionals related to the total variation and connections with image processing, Ann. PDE, Volume 4 (2018) no. 1, pp. 4-9

[7] A.M. Garsia; E. Rodemich Monotonicity of certain functionals under rearrangement, Ann. Inst. Fourier (Grenoble), Volume 24 (1974) no. 2, pp. 67-116

[8] H.-M. Nguyen Some new characterizations of Sobolev spaces, J. Funct. Anal., Volume 237 (2006) no. 2, pp. 689-720

[9] H.-M. Nguyen Γ-convergence and Sobolev norms, C. R. Acad. Sci. Paris, Ser. I, Volume 345 (2007) no. 12, pp. 679-684

[10] H.-M. Nguyen Further characterizations of Sobolev spaces, J. Eur. Math. Soc., Volume 10 (2008) no. 1, pp. 191-229

[11] H.-M. Nguyen Γ-convergence, Sobolev norms, and BV functions, Duke Math. J., Volume 157 (2011) no. 3, pp. 495-533

[12] H. Taylor Rearrangements of incidence tables, J. Comb. Theory, Ser. A, Volume 14 (1973), pp. 30-36

Cité par Sources :

Commentaires - Politique