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

Presented by: 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.

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.

Received:
Accepted:
Published online:
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 
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     title = {On the shape factor of interaction laws for a non-local approximation of the {Sobolev} norm and the total variation},
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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/

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[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

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