A general framework for nonlocal Neumann problems

TL;DR

This paper introduces a comprehensive Hilbert space framework for solving nonlocal Neumann problems, including fractional Laplace operators and peridynamics models, highlighting new theoretical insights and the transition to local boundary problems.

Contribution

It develops a general Hilbert space approach for nonlocal Neumann problems, extending existing theories to broader settings and linking nonlocal and local boundary value problems.

Findings

Established existence and uniqueness results for nonlocal Neumann problems.

Extended the framework to fractional Laplace operators and peridynamics models.

Demonstrated the transition from nonlocal to local boundary conditions.

Abstract

Within the framework of Hilbert spaces, we solve nonlocal problems in bounded domains with prescribed conditions on the complement of the domain. Our main focus is on the inhomogeneous Neumann problem in a rather general setting. We also study the transition from complement value problems to local boundary value problems. Several results are new even for the fractional Laplace operator. The setting also covers relevant models in the framework of peridynamics.