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We present a systematic study on a class of nonlocal integral functionals forfunctions defined on a bounded domain and the naturally induced functionspaces. The function spaces are equipped with a seminorm depending on finitedifferences weighted by a position-dependent function, which leads toheterogeneous localization on the domain boundary. We show the existence ofminimizers for nonlocal variational problems with classically-defined, localboundary constraints, together with the variational convergence of thesefunctionals to classical counterparts in the localization limit. This programnecessitates a thorough study of the nonlocal space; we demonstrate propertiessuch as a Meyers-Serrin theorem, trace inequalities, and compact embeddings,which are facilitated by new studies of boundary-localized convolutionoperators.

Nonlocal problems with local boundary conditions I: function spaces and variational principles