Characterization of function spaces of vector fields and an application in nonlinear peridynamics. (July 2016)

• Record Type: Journal Article
• Authors: Mengesha, Tadele; Du, Qiang
• Is Part Of: Nonlinear analysis. Volume 140(2016)
• Page Start: 82
• Page End: 111
• Publication Date: 2016-07
• Subjects: Nonlocal characterization -- Peridynamics -- Γ-convergence; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2016.02.024
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: The paper introduces a class of nonlocal derivative operators defined over vector fields that turn out to satisfy a nonlocal integration by parts formula. We demonstrate that in several function spaces, these operators have similar continuity property as the classical differential operators. A closed formula for the limit of these operators will be obtained when nonlocality vanishes. The limit analysis together with the integration by parts formula enables us to link the nonlocal derivative operators and associated function spaces with the conventional local differential operators and Sobolev, bounded variations, and bounded deformation function spaces. As an application, we present an existence result and asymptotic analysis in the sense of Γ -convergence of some nonlinear variational problems that arise in nonlocal continuum mechanics.