Nonlocal Differential Operators with Integrable, Nonsymmetric Kernels: Part I--Operator Theoretic Properties

TL;DR

This paper studies nonlocal differential operators with integrable, nonsymmetric kernels, establishing their properties, convergence to classical derivatives, and foundational results for nonlocal variational problems, serving as an accessible introduction to the field.

Contribution

It introduces a comprehensive analysis of nonsymmetric, integrable kernel-based nonlocal operators, including convergence, integration by parts, and compactness results, advancing the mathematical foundation for nonlocal modeling.

Findings

Operators converge to classical derivatives as nonlocality vanishes

New integration by parts formula for nonlocal operators

Characterization of operator compactness and implications for Poincaré inequality

Abstract

Recent decades have provided a host of examples and applications motivating the study of nonlocal differential operators. We discuss a class of such operators acting on bounded domains, focusing on those with integrable kernels having compact support. Notably, we make no explicit symmetry assumptions on the kernel and discuss some implications of this decision. We establish a nonlocal-to-local convergence result, showing that these operators coincide with the classical derivative as the nonlocality vanishes. We also provide a new integration by parts result, a characterization of the compactness of these nonlocal operators, an implication for the nonlocal Poincar\'e inequality, and a variety of examples. This work establishes several key results needed to analyze nonlocal variational problems, given in Part II. We hope that this paper can serve as a relatively gentle introduction to the field of nonlocal modeling.