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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 well-posedness 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 well-posedness
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 PoincarŽe-type 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, well-posedness, 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
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