Characterization of function spaces of vector fields and an application in nonlinear peridynamics

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 ΓΓ-convergence of some nonlinear variational problems that arise in nonlocal continuum mechanics.