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

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.