Mathematical analysis/Functional analysis
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.

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.

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.

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Published online:
DOI: 10.1016/j.crma.2018.05.014
Antonucci, Clara 1; Gobbino, Massimo 2; Migliorini, Matteo 1; Picenni, Nicola 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
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Antonucci, Clara; Gobbino, Massimo; Migliorini, Matteo; Picenni, Nicola. 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. http://www.numdam.org/articles/10.1016/j.crma.2018.05.014/

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

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

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

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

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

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

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

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

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

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

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