Nonlocal boundary-value problems with local boundary conditions

TL;DR

This paper develops a framework for nonlocal boundary-value problems with local boundary conditions, establishing well-posedness, a nonlocal Green's identity, and demonstrating convergence to classical solutions as the nonlocal interaction horizon shrinks.

Contribution

It introduces a heterogeneous localization approach for nonlocal operators with boundary conditions, proving well-posedness and convergence to classical solutions in the vanishing horizon limit.

Findings

Established a nonlocal Green's identity involving local boundary terms.

Proved well-posedness of variational formulations for nonlocal problems.

Demonstrated convergence of nonlocal solutions to classical solutions as horizon parameter approaches zero.

Abstract

We describe and analyze nonlocal integro-differential equations with classical local boundary conditions. The interaction kernel of the nonlocal operator has horizon parameter dependent on position in the domain, and vanishes as the boundary of the domain is approached. This heterogeneous localization allows for boundary values to be captured in the trace sense. We state and prove a nonlocal Green's identity for these nonlocal operators that involve local boundary terms. We use this identity to state and establish the well-posedness of variational formulations of the nonlocal problems with several types of classical boundary conditions. We show the consistency of these nonlocal boundary-value problems with their classical local counterparts in the vanishing horizon limit via the convergence of solutions. The Poisson data for the local boundary-value problem is permitted to be quite irregular, belonging to the dual of the classical Sobolev space. Heterogeneously mollifying this Poisson data for the local problem on the same length scale as the horizon and using the regularity of the interaction kernel, we show that the solutions to the nonlocal boundary-value problem with the mollified Poisson data actually belong to the classical Sobolev space, and converge weakly to the unique variational solution of the classical Poisson problem with original Poisson data.