A Nonlocal Vector Calculus with Application to Nonlocal Boundary Value Problems

Abstract. We develop a calculus for nonlocal operators that mimics Gauss’ theorem and the Green’s identities of the classical vector calculus. The operators we define do not involve the derivatives. We then apply the nonlocal calculus to define variational formulations of nonlocal “boundaryvalue” problems that mimic the Dirichlet and Neumann problems for second-order scalar elliptic partial differential equations. For the nonlocal variational problems, we derive fundamental solutions, show how one can derive existence and uniqueness results, and show how, under appropriate limits, they reduce to their classical analogs.