Nonlocal Boundary Conditions for Truncated-Fractional Differential Equations Posed on Bounded Domains

Abstract

Nonlocal models have found great success in recent years. In particular, models that replace spatial derivatives with strongly singular integral operators involving finite interaction radii are used in a wide range of applications, including image processing, continuum mechanics, and many other areas. However, from the perspective of mathematical analysis, difficulties arise when considering the appropriate notion of ``boundary'' conditions. (The nonlocal analogue of the boundary is often called a ``collar'' and is typically not a lower-dimensional set.) For instance, merely enforcing homogeneous Dirichlet-type constraints on the collar fails to guarantee a number of essential properties used in the analysis and numerical simulation of these models, as we show in the present work. In the local setting, working in a Sobolev space of functions that are zero on the boundary allows one to integrate by parts with no boundary term, approximate by test functions, extend by zero to the full space, and apply the Hardy and Poincaré inequalities. We prove that the nonlocal analog of each of these fails if one only requires the functions to be zero on the collar, and that one must move to a smaller, more restrictive space to enjoy these properties. In addition, we introduce a new lifting operator, correct an error in a previously published result on a nonlocal Green's theorem, and establish how the properties listed above relate to existing notions of fractional Sobolev spaces.