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We consider a family of non-local and non-convex functionals, and we provethat their Gamma-liminf is bounded from below by a positive multiple of theSobolev norm or the total variation. As a by-product, we answer some openquestions concerning the limiting behavior of these functionals.  The proof relies on the analysis of a discretized version of thesefunctionals.

Gamma-liminf estimate for a class of non-local approximations of Sobolev and BV norms