We consider the family of non-local and non-convex 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 one.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 one, and then to a finite combinatorial rearrangement inequality.Mathematics Subject Classification 2010 (MSC2010): 26B30, 46E35.

We consider the approximation of the total variation of a function by the family of non-local and non-convex functionals introduced by H. Brezis and H.-M. Nguyen in a recent paper. The approximating functionals are defined through double integrals in which every pair of points contributes according to some interaction law.In this paper we answer two open questions concerning the dependence of the Gamma-limit on the interaction law. In the first result, we show that the Gammalimit depends on the full shape of the interaction law, and not only on the values in a neighborhood of the origin. In the second result, we show that there do exist interaction laws for which the Gamma-limit coincides with the pointwise limit on smooth functions.The key argument is that for some special classes of interaction laws the computation of the Gamma-limit can be reduced to studying the asymptotic behavior of suitable multi-variable minimum problems.Mathematics Subject Classification 2010 (MSC2010): 26B30, 46E35.

We consider the family of non-local and non-convex functionals introduced in a recent paper by H. Brezis and H.-M. Nguyen. 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.Mathematics Subject Classification 2010 (MSC2010): 26B30, 46E35.

We provide a counterexample to an open question concerning a characterization of constant functions through double integrals that involve different quotients. This counterexample requires the construction of an unbounded function whose difference quotients avoid a sequence of intervals with endpoints that diverge to infinity.