We consider the family of nonlocal and nonconvex functionals proposed
and investigated by J. Bourgain, H. Brezis and H.-M. Nguyen in a series
of papers of the last decade. It was known that this family of functionals
Gamma-converges to a suitable multiple of the Sobolev norm or the total
variation, depending on the summability exponent, but the exact constants and
the structure of recovery families were still unknown, even in dimension
1.
We prove a Gamma-convergence result with explicit values of the constants in any
space dimension. We also show the existence of recovery families consisting of smooth
functions with compact support.
The key point is reducing the problem first to dimension 1, and then to a finite
combinatorial rearrangement inequality.