Nonlocal problems with local boundary conditions I: function spaces and variational principles
Abstract
We present a systematic study on 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. We show 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; we demonstrate properties such as a Meyers-Serrin theorem, trace inequalities, and compact embeddings, which are facilitated by new studies of boundary-localized convolution operators.
Cite
@article{arxiv.2307.08855,
title = {Nonlocal problems with local boundary conditions I: function spaces and variational principles},
author = {James M. Scott and Qiang Du},
journal= {arXiv preprint arXiv:2307.08855},
year = {2023}
}