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

TL;DR

This paper develops a comprehensive framework for nonlocal variational problems with local boundary conditions, introducing new function spaces and demonstrating convergence to classical models, with key properties like compactness and trace inequalities.

Contribution

It introduces a novel class of nonlocal function spaces with heterogeneous boundary localization and proves their fundamental properties, including existence of minimizers and convergence to classical functionals.

Findings

Existence of minimizers for nonlocal variational problems.

Demonstration of variational convergence to classical models.

Establishment of properties like trace inequalities and compact embeddings.

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.