Nonlocal problems with local boundary conditions II: Green's identities and regularity of solutions

TL;DR

This paper develops Green's identities for nonlocal integral equations with boundary-localized interactions, enabling analysis of boundary conditions, well-posedness, and regularity of solutions, bridging nonlocal and classical PDE theories.

Contribution

It introduces a Green's identity for nonlocal operators with boundary localization, facilitating the analysis of boundary conditions and regularity of solutions.

Findings

Established Green's identity for nonlocal operators

Proved well-posedness of nonlocal boundary value problems

Demonstrated Sobolev convergence to classical solutions

Abstract

We study nonlocal integral equations on bounded domains with finite-range nonlocal interactions that are localized at the boundary. We establish a Green's identity for the nonlocal operator that recovers the classical boundary integral, which, along with the variational analysis established previously, leads to the well-posedness of these nonlocal problems with various types of classical local boundary conditions. We continue our analysis via boundary-localized convolutions, using them to analyze the Euler-Lagrange equations, which permits us to establish global regularity properties and classical Sobolev convergence to their classical local counterparts.