Calculus of Variations and Geometric Measure Theory
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D. Spector - T. Mengesha

Localization of Nonlocal Gradients in Various Topologies

created by spector on 24 Jul 2013
modified on 05 Feb 2014

[BibTeX]

Accepted Paper

Inserted: 24 jul 2013
Last Updated: 5 feb 2014

Journal: Calc. Var. and PDE
Year: 2013

Abstract:

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


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