Non-constant functions with zero nonlocal gradient and their role in nonlocal Neumann-type problems

TL;DR

This paper studies functions with zero nonlocal gradient, revealing their structure, regularity, and applications to nonlocal PDEs, including boundary value problems and Neumann conditions, with implications for nonlocal Sobolev spaces.

Contribution

It characterizes functions with zero nonlocal gradient, introduces new tools for nonlocal Sobolev spaces, and analyzes nonlocal PDEs with natural boundary conditions.

Findings

Functions with zero nonlocal gradient form an infinite-dimensional vector space.

New nonlocal Poincaré inequalities and compactness results are established.

Well-posedness and local limit links for nonlocal Neumann problems are proved.

Abstract

This work revolves around properties and applications of functions whose nonlocal gradient, or more precisely, finite-horizon fractional gradient, vanishes. Surprisingly, in contrast to the classical local theory, we show that this class forms an infinite-dimensional vector space. Our main result characterizes the functions with zero nonlocal gradient in terms of two simple features, namely, their values in a layer around the boundary and their average. The proof exploits recent progress in the solution theory of boundary-value problems with pseudo-differential operators. We complement these findings with a discussion of the regularity properties of such functions and give illustrative examples. Regarding applications, we provide several useful technical tools for working with nonlocal Sobolev spaces when the common complementary-value conditions are dropped. Among these, are new nonlocal Poincar\'e inequalities and compactness statements, which are obtained after factoring out functions with vanishing nonlocal gradient. Following a variational approach, we exploit the previous findings to study a class of nonlocal partial differential equations subject to natural boundary conditions, in particular, nonlocal Neumann-type problems. Our analysis includes a proof of well-posedness and a rigorous link with their classical local counterparts via $\Gamma$-convergence as the fractional parameter tends to 1.