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We study nonlocal integral equations on bounded domains with finite-rangenonlocal interactions that are localized at the boundary. We establish aGreen's identity for the nonlocal operator that recovers the classical boundaryintegral, which, along with the variational analysis established previously,leads to the well-posedness of these nonlocal problems with various types ofclassical local boundary conditions. We continue our analysis viaboundary-localized convolutions, using them to analyze the Euler-Lagrangeequations, which permits us to establish global regularity properties andclassical Sobolev convergence to their classical local counterparts.

Nonlocal problems with local boundary conditions II: Green's identities and regularity of solutions