  We describe and analyze nonlocal integro-differential equations with
classical local boundary conditions. The interaction kernel of the nonlocal
operator has horizon parameter dependent on position in the domain, and
vanishes as the boundary of the domain is approached. This heterogeneous
localization allows for boundary values to be captured in the trace sense. We
state and prove a nonlocal Green's identity for these nonlocal operators that
involve local boundary terms. We use this identity to state and establish the
well-posedness of variational formulations of the nonlocal problems with
several types of classical boundary conditions.
  We show the consistency of these nonlocal boundary-value problems with their
classical local counterparts in the vanishing horizon limit via the convergence
of solutions. The Poisson data for the local boundary-value problem is
permitted to be quite irregular, belonging to the dual of the classical Sobolev
space. Heterogeneously mollifying this Poisson data for the local problem on
the same length scale as the horizon and using the regularity of the
interaction kernel, we show that the solutions to the nonlocal boundary-value
problem with the mollified Poisson data actually belong to the classical
Sobolev space, and converge weakly to the unique variational solution of the
classical Poisson problem with original Poisson data.
