Nonlocal problems with local boundary conditions. I: Function spaces and variational principles (Q6571027)

The authors systematically study a class of nonlocal integral functionals for functions defined on a bounded domain and the naturally induced function spaces. The function spaces are equipped with a seminorm depending on finite differences weighted by a position-dependent function, which leads to heterogeneous localization on the domain boundary. The authors prove the existence of minimizers for nonlocal variational problems with classically defined, local boundary constraints, together with the variational convergence of these functionals to classical counterparts in the localization limit. This program necessitates a thorough study of the nonlocal space; the properties such as a Meyers-Serrin theorem, trace inequalities, and compact embeddings are also demonstrated, which are facilitated by new studies of boundary-localized convolution operators. The results in this paper are very interesting and significant.