5 research outputs found
Nonlocal gradient operators with a nonspherical interaction neighborhood and their applications
Nonlocal gradient operators are prototypical nonlocal differential operators
thatare very important in the studies of nonlocal models. One of the simplest
variational settings for such studies is the nonlocal Dirichlet energies
wherein the energy densities are quadratic in the nonlocal gradients. There
have been earlier studies to illuminate the link between the coercivity of the
Dirichlet energies and the interaction strengths of radially symmetric kernels
that constitute nonlocal gradient operators in the form of integral operators.
In this work we adopt a different perspective and focus on nonlocal gradient
operators with a non-spherical interaction neighborhood. We show that the
truncation of the spherical interaction neighborhood to a half sphere helps
making nonlocal gradient operators well-defined and the associated nonlocal
Dirichlet energies coercive. These become possible, unlike the case with full
spherical neighborhoods, without any extra assumption on the strengths of the
kernels near the origin. We then present some applications of the nonlocal
gradient operators with non-spherical interaction neighborhoods. These include
nonlocal linear models in mechanics such as nonlocal isotropic linear
elasticity and nonlocal Stokes equations, and a nonlocal extension of the
Helmholtz decomposition.Comment: Corrected typo