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arXiv · 2307.08855

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

James M. Scott·Qiang Du
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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.

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James M. Scott, Qiang Du. 2023-07-17. Nonlocal problems with local boundary conditions I: function spaces and variational principles. https://arxiv.org/abs/2307.08855

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