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

Carolin Kreisbeck, Hidde Schönberger

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 Poincaré 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 $Γ$-convergence as the fractional parameter tends to 1.

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

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 Poincaré 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 -convergence as the fractional parameter tends to 1.
Paper Structure (19 sections, 25 theorems, 192 equations, 3 figures)

This paper contains 19 sections, 25 theorems, 192 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Nonlocal gradients and Sobolev spaces
  5. Translation operators
  6. Pseudo-differential operators and Dirichlet problems
  7. Discussion and characterization of functions with zero nonlocal gradient
  8. Non-constant elements of $N^{s, p, \delta}(\Omega)$
  9. Characterization of $N^{s, p, \delta}(\Omega)$
  10. Regularity properties of functions with zero nonlocal gradient and examples
  11. Technical tools involving functions with zero nonlocal gradient
  12. Connection between classical and nonlocal Sobolev spaces
  13. Extension modulo functions with zero nonlocal gradient.
  14. A new nonlocal Poincaré inequality
  15. Nonlocal Poincaré-Wirtinger inequality
  16. ...and 4 more sections

Key Result

Lemma 2.4

Let $s\in (0,1)$, $\delta>0$, $p \in [1,\infty]$, and $\Omega \subset \mathbb{R}^n$ be open. Then, the linear map $\mathcal{Q}_\delta^s: H^{s, p, \delta}(\Omega)\to W^{1,p}(\Omega), \ u\mapsto Q_\delta^s\ast u$ is bounded (uniformly with respect to $s$) with

Figures (3)

  • Figure 1: Illustration of a set $\Omega\subset \mathbb{R}^n$ with its expansion $\Omega_\delta$, the outer and inner collar regions $\Gamma_\delta$ (green) and $\Gamma_{-\delta}$ (light green), and the reduced set $\Omega_{-\delta}$ (gray).
  • Figure 2: Numerical approximation of the function $h_{c,g} \in N^{s,2,\delta}(\Omega)$ for $c=0$ and $g\equiv -1$ on $\Gamma_\delta$ with increasing degrees of zoom. The parameters for the computation are $n=1$, $\Omega=(-3,3)$, $s=\frac{1}{2}$, $\delta=1$ and $w_\delta \in C_c^{\infty}(-1,1)$ is a bump function equal to $1$ on $(-\frac{1}{2},\frac{1}{2})$.
  • Figure 3: Left: A numerical approximation of the function $h_{c,g} \in N^{s,2,\delta}(\Omega)$ with $c=0$ and $g(x)=x$ for $x \in \Gamma_\delta$. Right: A plot of $\mathcal{P}^s_\delta v|_{\Omega_\delta} \in \bigcap_{p \in [1,\infty]}N^{s, p, \delta}(\Omega)$ with $v(x)=1+5\varphi(x+4)-2\varphi(x-4)$ for a non-negative bump function $\varphi \in C_c^{\infty}(-1,1)$. The parameters are the same as in Figure \ref{['fig:zoom']}.

Theorems & Definitions (62)

  • Remark 2.1
  • Definition 2.2: Nonlocal Sobolev spaces
  • Remark 2.3
  • Lemma 2.4: From nonlocal to local gradients
  • Lemma 2.5: From local to nonlocal gradients
  • Theorem 2.6: Existence and uniqueness for pseudo-differential Dirichlet problems
  • proof
  • Remark 2.7
  • Lemma 2.8: $\mathcal{P}_\delta^s$ as pseudo-differential operator
  • proof
  • ...and 52 more