  We consider linear and obstacle problems driven by a nonlocal integral
operator, for which nonlocal interactions are restricted to a ball of finite
radius. These type of operators are used to model anomalous diffusion and, for
a special choice of the integral kernels, reduce to the fractional Laplace
operator on a bounded domain. By means of a nonlocal vector calculus we recast
the problems in a weak form, leading to corresponding nonlocal variational
equality and inequality problems. We prove optimal regularity results for both
problems, including a higher regularity of the solution and the Lagrange
multiplier. Based on the regularity results, we analyze the convergence of
finite element approximations for a linear problem and illustrate the
theoretical findings by numerical results.
