Inserted: 24 jul 2013
Last Updated: 5 feb 2014
Journal: Calc. Var. and PDE
In this paper, we study nonlocal gradients and their relationship to classical gradients. As the nonlocality vanishes we demonstrate the convergence of nonlocal gradients to their local analogue for Sobolev and BV functions. As a consequence of these localizations we give new characterizations of the Sobolev and BV spaces that are in the same spirit of Bourgain, Brezis, and Mironsecu's 2001 characterization. Integral functionals of the nonlocal gradient with proper growth are shown to converge to a corresponding functional of the classical gradient both pointwise and in the sense of Gamma-convergence.
Keywords: Sobolev spaces, BV Spaces, Nonlocal Gradients