TY - JOUR
T1 - Localization of nonlocal gradients in various topologies
AU - Mengesha, Tadele
AU - Spector, Daniel
N1 - Funding Information:
Tadele Mengesha would like to thank Prof. Qiang Du of Penn State University for his encouragement and constant support. His research is supported in part by the Dept. of Energy grant DE-SC0005346 and NSF Grant DMS-1312809. Daniel Spector would like to thank Prof. Itai Shafrir, without whom this paper certainly would not have been possible. His research is supported in part by a Technion Fellowship. The authors would also like to thank the anonymous referee for his or her valuable comments in improving the paper.
Publisher Copyright:
© Springer-Verlag Berlin Heidelberg 2014.
PY - 2015/1
Y1 - 2015/1
N2 - 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.
AB - 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.
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U2 - 10.1007/s00526-014-0711-3
DO - 10.1007/s00526-014-0711-3
M3 - Article
AN - SCOPUS:84924521939
VL - 52
SP - 253
EP - 279
JO - Calculus of Variations and Partial Differential Equations
JF - Calculus of Variations and Partial Differential Equations
SN - 0944-2669
IS - 1-2
ER -