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

Antonucci, Clara1 ; Gobbino, Massimo2 ; Migliorini, Matteo1 ; Picenni, Nicola1
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, Tome 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
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, Tome 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/

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