 
Summary: RESULTS ON NONLOCAL BOUNDARY VALUE PROBLEMS
BURAK AKSOYLU AND TADELE MENGESHA
Abstract. In this article, we provide a variational theory for nonlocal problems where nonlocality
arises due to interaction in a given horizon. With this theory, we prove wellposedness results for
the weak formulation of nonlocal boundary value problems with Dirichlet, Neumann, and mixed
boundary conditions for a class of kernel functions. The motivating application for nonlocal bound
ary value problems is the scalar stationary peridynamics equation of motion. The wellposedness
results support practical kernel functions used in the peridynamics setting.
We also prove a spectral equivalence estimate which leads to a mesh size independent upper
bound for the condition number of an underlying discretized operator. This is a fundamental con
ditioning result that would guide preconditioner construction for nonlocal problems. The estimate
is a consequence of a nonlocal PoincarŽetype inequality that reveals a horizon size quantification.
We provide an example that establishes the sharpness of the upper bound in the spectral equiva
lence.
Keywords: Nonlocal operators, nonlocal boundary value problems, wellposedness, nonlocal PoincarŽe
inequality, peridynamics, condition number, preconditioning.
1. Introduction
Nonlocal problems have become a critical part of modeling and simulation of complex phenomena
that span vastly different length scales. Examples include evolution equations for species population
densities [8], image processing [14, 24], porous media flow [9, 10, 22], and turbulence [4]. The book
