Nonlocal gradient operators with a nonspherical interaction neighborhood and their applications

Nonlocal gradient operators are prototypical nonlocal differential operators that are 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.