Localization of nonlocal gradients in various topologies

Tadele Mengesha, Daniel Spector*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

18 Citations (Scopus)

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 (Optimal control and partial differential equations (a volume in honour of A. Benssoussan’s 60th birthday). IOS Press, Amsterdam, pp. 439–455. 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 Γ-convergence.

Original languageEnglish
Pages (from-to)253-279
Number of pages27
JournalCalculus of Variations and Partial Differential Equations
Volume52
Issue number1-2
DOIs
Publication statusPublished - 2015 Jan
Externally publishedYes

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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