# Nonlocal diffusion models with consistent local and fractional
limits††thanks: This work is to be published in Approximation, Applications,
and Analysis of Nonlocal, Nonlinear Models (The 50th John H. Barrett Memorial
Lectures), 2022. The research of Q. Du was supported in part by NSF grant
DMS-2012562 and DMS-1937254. The research of X. Tian was supported in part by
NSF grant DMS-2111608. The research of Z. Zhou is partially supported by Hong
Kong Research Grants Council (No. 15303122) and an internal grant of Hong Kong
Polytechnic University (Project ID: P0031041, Programme: ZZKS)

Qiang Du Department of Applied Physics and Applied Mathematics, and Data
Science Institute, Columbia University, New York, NY 10027, USA
(qd2125@columbia.edu). Xiaochuan Tian Department of Mathematics, University of
California, San Diego, CA 92093, USA (xctian@ucsd.edu). Zhi Zhou Department of
Applied Mathematics. Hong Kong Polytechnic University, Kowloon, Hong Kong,
China (zhizhou@polyu.edu.hk).

###### Abstract

For some spatially nonlocal diffusion models with a finite range of nonlocal
interactions measured by a positive parameter $\delta$, we review their
formulation defined on a bounded domain subject to various conditions that
correspond to some inhomogeneous data. We consider their consistency to
similar inhomogeneous boundary value problems of classical partial
differential equation (PDE) models as the nonlocal interaction kernel gets
localized in the local $\delta\to 0$ limit, and at the same time, for rescaled
fractional type kernels, to corresponding inhomogeneous nonlocal boundary
value problems of fractional equations in the global $\delta\to\infty$ limit.
Such discussions help to delineate issues related to nonlocal problems defined
on a bounded domain with inhomogeneous data.

###### keywords:

nonlocal models, nonlocal diffusion, peridynamics, fractional PDEs, well-
posedness, inhomogeneous data

###### AMS:

47G10, 46E35, 35R11,35J25

##  1 Introduction

There has been much interest in nonlocal models [4, 10, 28, 20, 46]. A
characterization of nonlocal models can be found in [29], together with a
study on the close connections between nonlocal modeling and other
mathematical subjects like homogenization, model reduction, and coarse-
graining.

Motivated by studies from mechanics to image analysis and from traffic flows
of autonomous vehicles to anomalous diffusion, a growing area of research is
the study on models with nonlocal interactions that span a finite range [28,
30], measured by the horizon parameter $\delta\in\mathbb{R}$, as we will
describe in details later. It should be noted that in such models, the
interaction range is generically nonzero (thus nonlocal) but, for problems on
a bounded domain, may not be on the same scale as the size of the whole
domain. In mathematical modeling, nonlocal models with a finite $\delta$,
simply referred to as nonlocal models here, can play a number of roles [28]:

  * •

They can either complement or serve as an alternative to traditional PDE-based
local models, and they can account for nonlocal interactions explicitly and
remain valid for not only smooth but also singular solutions. Examples include
peridynamic (PD) models for fracture mechanics [65, 36] and nonlocal models of
anomalous diffusion and transport [30, 56, 57, 4].

  * •

They can serve as tools to better understand and simulate existing local PDE
models. Popular numerical methods like Smoothed Particle Hydrodynamics (SPH)
and Vortex-Blob methods are good examples of discretizing PDEs through
nonlocal smoothing/averaging [18, 58, 37, 35].

  * •

They can serve as bridges to build connections between continuum PDEs,
discrete algebraic, graph and network models, metric geometry, and fractional
differential equations [11, 16, 15, 27, 74, 75, 28].

The main goal here is related to the bridging roles mentioned above. That is,
the discussion is centered on connecting local PDEs with fractional PDEs
through nonlocal equations with a finite range of interactions via a unified
description. In particular, we reexamine some concepts presented in [30] on
nonlocal diffusion models with a finite $\delta$, and make further
investigations so that the theory can be consistent with both the local and
fractional limits. While the related issues have been subject to earlier
investigations, see e.g. [30, 75, 20], most known results are stated for
problems with homogeneous boundary data (or nonlocal constraints). By
presenting problems in more general settings that involve inhomogeneous data,
we also provide some insight on how to connect some of the relevant concepts
concerning nonlocal boundary conditions (volumetric constraints [30]).

##  2 Local, nonlocal and fractional models

For illustration, we consider problems defined on a bounded, open domain
${\Omega}\subset{{\mathbb{R}}^{d}}$ with a Lipshitz boundary
${\partial{\Omega}}$. We adopt a set of notation that is largely similar to
those used in [20] but with necessary modifications in some cases.

Let the closure of ${\Omega}$ be denoted by
${\overline{{\Omega}}}={\Omega}\cup\partial{\Omega}$. The complements are
denoted by ${{\Omega}_{c}}={{\mathbb{R}}^{d}}\setminus{\overline{{\Omega}}}$
and ${{\overline{{\Omega}}}_{c}}={{\mathbb{R}}^{d}}\setminus{\Omega}$
respectively. Note that
${{\overline{{\Omega}}}_{c}}=\overline{{{\mathbb{R}}^{d}}\setminus{\Omega}}=\overline{{{\Omega}_{c}}}$.

Given the domain ${\Omega}$ and a positive parameter $\delta>0$, we define an
interaction layer of width $\delta$ by

${{\Omega}_{{\mathcal{I}}_{\delta}}}:=\\{{\bm{y}}\in{{\Omega}_{c}}\;\mid\;\text{dist}({\bm{y}},{\Omega})<\delta\,\\}.$
(1)

We let $\overline{{{\Omega}_{{\mathcal{I}}_{\delta}}}}$ denote the closure of
${{\Omega}_{{\mathcal{I}}_{\delta}}}$.
${{\widehat{{\Omega}}}_{\delta}}={\Omega}\cup{{\Omega}_{{\mathcal{I}}_{\delta}}}$
and
${{\overline{{\Omega}}}_{{\delta}}}=\overline{{\Omega}\cup{{\Omega}_{{\mathcal{I}}_{\delta}}}}={\overline{{\Omega}}}\cup\overline{{{\Omega}_{{\mathcal{I}}_{\delta}}}}$.
In this work, we are interested in some special cases and limiting regimes

$\delta=0,\quad
0\leftarrow{\delta}\ll\text{diam\,}{\Omega},\quad{\delta}\approx\text{diam\,}{\Omega},\quad\text{diam\,}{\Omega}\ll{\delta}\to\infty,\quad{\delta}=\infty.$
(2)

Note that ${{\Omega}_{{\mathcal{I}}_{0}}}=\emptyset$ and
$\overline{{{\Omega}_{{\mathcal{I}}_{0}}}}={\partial{\Omega}}$, while
${{\Omega}_{{\mathcal{I}}_{\infty}}}={{\Omega}_{c}}$ and
$\overline{{{\Omega}_{{\mathcal{I}}_{\infty}}}}={{\overline{{\Omega}}}_{c}}$.

We also define some domains for any pair of points ${({\bm{x}},{\bm{y}})}$.
These include ${\Omega}^{2}={\Omega}\times{\Omega}$, which denotes the tensor
product of any domain ${\Omega}$ with itself and

${\widetilde{{\Omega}}_{\delta}}={{{\widehat{{\Omega}}}_{\delta}}^{2}\setminus{\Omega}^{2}_{{\mathcal{I}}_{\delta}}}\quad\text{and}\quad{\widetilde{{\Omega}}_{\infty}}=({{\mathbb{R}}^{d}})^{2}\setminus{\Omega}_{c}^{2}.$

###  2.1 Local diffusion models

We first consider a classical Poisson problem for an unknown solution
$u\colon\Omega\subset{{\mathbb{R}}^{d}}\to\mathbb{R}$,

$\begin{cases}-{{\mathcal{L}}_{0}}u({\bm{x}}):=-(\Delta
u)({\bm{x}})=F({\bm{x}},u({\bm{x}})),&\quad\forall\,{\bm{x}}\in{\Omega},\\\
{{\mathcal{B}}_{0}}u({\bm{x}})=g_{0}({\bm{x}}),&\quad\forall\,{\bm{x}}\in\overline{{{\Omega}_{{\mathcal{I}}_{0}}}}={\partial{\Omega}},\end{cases}$
(3)

where $F({\bm{x}},u)$ and $g_{0}({\bm{x}})$ denote given functions defined on
${\Omega}\times{\mathbb{R}}$ and $\partial{\Omega}$, respectively. Here, we
elect to work with the Laplacian $\Delta$ rather than more general diffusion
operators. A special case is that $F({\bm{x}},u)$ is independent of $u$, thus
leading to a standard linear PDE on ${\Omega}$. For the boundary conditions
operator ${{\mathcal{B}}_{0}}$, we have the choices

${{\mathcal{B}}_{0}}u({\bm{x}})=\begin{cases}u({\bm{x}}),&{\bm{x}}\in\partial{\Omega}^{D}\;\mbox{(Dirichlet
BC)},\\\ \displaystyle\frac{\partial
u}{\partial\mathbf{n}}({\bm{x}})&{\bm{x}}\in\partial{\Omega}^{N}\;\mbox{(Neumann
BC)},\end{cases}$ (4)

where ${\partial{\Omega}^{D}}$ and ${\partial{\Omega}^{N}}$ are subset of
${\partial{\Omega}}$ such that
${\partial{\Omega}^{D}}\cup{\partial{\Omega}^{N}}={\partial{\Omega}}$ with
${\partial{\Omega}^{D}}\cap{\partial{\Omega}^{N}}$ having zero $d-1$
dimensional measure. Note that ${{\mathcal{B}}_{0}}$ can also take on other
forms, (e.g.,

${{\mathcal{B}}_{0}}u({\bm{x}})=\displaystyle\frac{\partial
u}{\partial\mathbf{n}}({\bm{x}})-\alpha
u({\bm{x}}),\quad\forall{\bm{x}}\in\partial{\Omega}^{N}$ (5)

for a positive constant $\alpha>0$) to yield other (e.g., Robin type) boundary
conditions. In the case ${\partial{\Omega}^{D}}={\partial{\Omega}}$, we have a
pure Dirichlet problem while ${\partial{\Omega}^{N}}={\partial{\Omega}}$ leads
to a pure Neumann problem. In the latter case, compatibility conditions may be
needed on the data, and an additional constraint (e.g., having a zero mean
over ${\Omega}$) on the solutions may be imposed to ensure uniqueness.

######  Remark 2.1.

Problem (3), which is named the local diffusion problem here, is a well-
studied local PDE model used for steady-state normal diffusion.

###  2.2 Fractional diffusion models

Fractional PDEs are PDEs with fractional-order partial derivatives of unknown
solutions. They are nonlocal integral models with a global or infinite range
of interactions. Here, we consider a nonlocal fractional diffusion model
associated with the integral fractional Laplacian
${{\mathcal{L}}_{\infty}^{s}}=-(-\Delta)^{s}$, which, for
$u\colon{{\mathbb{R}}^{d}}\to\mathbb{R}$ and $s\in(0,1)$, is given by

$\displaystyle-{{\mathcal{L}}_{\infty}^{s}}u({\bm{x}})=(-\Delta)^{s}u({{\bm{x}}})$
$\displaystyle=\int_{{{\mathbb{R}}^{d}}}(u({{\bm{x}}})-u({{\bm{y}}})){{\gamma}_{s}}{(|{{\bm{y}}-{\bm{x}}}|)}{d{\bm{y}}},\quad\quad\forall\,{\bm{x}}\in{{\mathbb{R}}^{d}}$
(6)

where the kernel function is defined by

${{\gamma}_{s}}(|{\bm{z}}|)=\displaystyle\frac{{C_{{d},s}}}{|{\bm{z}}|^{{d}+2s}},\quad\text{with}\quad{C_{{d},s}}=\displaystyle\frac{2^{2s}s\Gamma(s+{{{d}}/{2}})}{\pi^{{d}/2}\Gamma(1-s)},$
(7)

where $\Gamma$ denotes the gamma function. Note that the singular integral in
(6) should be interpreted in the principal value sense.

The _fractional diffusion problem_ considered here is then given by

$\begin{cases}-{{\mathcal{L}}_{\infty}^{s}}u({\bm{x}})=F({\bm{x}},u({\bm{x}})),&\quad\forall\,{\bm{x}}\in{\Omega}{,}\\\
{{\mathcal{B}}_{\infty}^{s}}u({\bm{x}})=g_{\infty}({\bm{x}}),&\quad\forall\,{\bm{x}}\in{{{\Omega}_{{\mathcal{I}}_{\infty}}}}={{\Omega}_{c}},\end{cases}$
(8)

where $F({\bm{x}},u)$ is as before and $g_{\infty}({\bm{x}})$ is a given data
defined for ${\bm{x}}\in{{\Omega}_{{\mathcal{I}}_{\infty}}}$. The operator
${{\mathcal{B}}_{\infty}^{s}}$ is defined by

${{\mathcal{B}}_{\infty}^{s}}u({\bm{x}})=\begin{cases}u({\bm{x}}),&{\bm{x}}\in{{\Omega}^{D}_{{\mathcal{I}}_{\infty}}}\;\mbox{(Dirichlet
VC)},\\\
{{\mathcal{N}}_{\infty}^{s}}u({\bm{x}}),&{\bm{x}}\in{{\Omega}^{N}_{{\mathcal{I}}_{\infty}}}\;\mbox{(Neumann
VC)},\end{cases}$ (9)

where ${{\Omega}^{D}_{{\mathcal{I}}_{\infty}}}$ and
${{\Omega}^{N}_{{\mathcal{I}}_{\infty}}}$ are two disjoint subdomains of
${{\Omega}_{{\mathcal{I}}_{\infty}}}$ such that
$\overline{{{\Omega}^{D}_{{\mathcal{I}}_{\infty}}}\cup{{\Omega}^{N}_{{\mathcal{I}}_{\infty}}}}={{\overline{{\Omega}}}_{c}}$.
${{\mathcal{N}}_{\infty}^{s}}$ is a suitably defined nonlocal operator, for
example, given by [26],

${{\mathcal{N}}_{\infty}^{s}}u({{\bm{x}}})=-\int_{{\Omega}}(u({{\bm{y}}})-u({{\bm{x}}})){{\gamma}_{s}}{(|{{\bm{y}}-{\bm{x}}}|)}{{d{\bm{y}}}},\quad\quad\forall\,{\bm{x}}\in{{{\Omega}_{{\mathcal{I}}_{\infty}}}={{\Omega}_{c}}}.$
(10)

In the above definition, ${{\mathcal{N}}_{\infty}^{s}}$ is defined as an
integral over the domain ${\Omega}$ only, instead of the whole space
${{\mathbb{R}}^{d}}$.

######  Remark 2.2.

Note that the constraint ${{\mathcal{B}}_{\infty}^{s}}u=g_{\infty}$ is applied
on a domain having a nonzero volume in ${{\mathbb{R}}^{d}}$, thus representing
a nonlocal boundary condition which is also called a volume constraint (VC)
[30].

The integral fractional Laplacian is sometimes called the whole space
fractional Laplacian as the integral ranges the whole space
${{\mathbb{R}}^{d}}$. It is a special case of the nonlocal Laplacian defined
later (see (19) and related discussions in section 2.3). It is also equivalent
to the Fourier representation [76]

$\displaystyle(-\Delta)^{s}u({\bm{x}})$
$\displaystyle=\displaystyle\frac{1}{(2\pi)^{d}}\int_{{{\mathbb{R}}^{d}}}|{\boldsymbol{\xi}}|^{2s}(u,\mathrm{e}^{-{\mathrm{i}}{\boldsymbol{\xi}}\cdot{\bm{x}}})\,\mathrm{e}^{{\mathrm{i}}{\boldsymbol{\xi}}\cdot{\bm{x}}}{d}{\boldsymbol{\xi}}=\mathcal{F}^{-1}(|{\boldsymbol{\xi}}|^{2s}\mathcal{F}\\{u\\}({\boldsymbol{\xi}}))({\bm{x}}),$
(11)

where $\mathcal{F}$ denotes the Fourier transform. Formally if we let $s\to
1^{-}$ in the above, it recovers the Fourier spectral representation of the
classical local Laplacian $-\Delta$. Indeed, it is well known that
$(-\Delta)^{s}u(x)\rightarrow-\Delta u(x)$ as $s\to 1^{-}$ for any function
$u$ with sufficient regularity; see e.g. [68, Theorems 3 and 4] and [25,
Proposition 4.4].

######  Remark 2.3.

The fractional Laplacian (6) is often used to model superdiffusion. In
probabilistic terms, in contrast to the local Laplacian that corresponds to
the normal diffusion described by Brownian motion, the fractional Laplacian
can be connected to superdiffusion associated with Lévy flights in which the
length of particle jumps follows a heavy-tailed power law distribution,
reflecting the long-range interactions between particles, see e.g. [26, 56,
66].

The problem (8) is called a fractional diffusion model here to distinguish it
from other diffusion models. It is given in the distribution sense so that the
equations are only specified over an open domain ${\Omega}$. More rigorous
descriptions of (8) will be discussed later.

Other variants of the operators ${{\mathcal{L}}_{\infty}^{s}}$ and
${{\mathcal{N}}_{\infty}^{s}}$ are discussed in section 2.4. For notation
simplicity, we drop the superscript $s$ in ${{\mathcal{L}}_{\infty}^{s}}$,
${{\mathcal{B}}_{\infty}^{s}}$ and ${{\mathcal{N}}_{\infty}^{s}}$ whenever
there is no ambiguity, that is
${{\mathcal{L}}_{\infty}^{s}}={{\mathcal{L}}_{\infty}}$,
${{\mathcal{B}}_{\infty}^{s}}={{\mathcal{B}}_{\infty}}$ and
${{\mathcal{N}}_{\infty}^{s}}={{\mathcal{N}}_{\infty}}$. Meanwhile, similar to
the local case, one can also consider other volume constraints such as
${{\mathcal{B}}_{\infty}}u({\bm{x}})={{\mathcal{N}}_{\infty}}u({\bm{x}})-\alpha
u({\bm{x}})$ for ${\bm{x}}\in{{\Omega}_{c}}$ to mimic a Robin type constraint.

###  2.3 Nonlocal diffusion models

To distinguish from the local and fractional counterparts, we consider a class
of special nonlocal diffusion models here associated with a finite range of
interactions, measured by the so-called horizon parameter (interaction radius)
$\delta>0$.

For ${\delta}>0$, we consider the following nonlocal problem for a scalar-
valued unknown function $u({\bm{x}})$ defined on
${{\widehat{{\Omega}}}_{\delta}}$,

$\begin{cases}-{{\mathcal{L}}_{\delta}}u=F({\bm{x}},u({\bm{x}})),&\quad\forall\,{\bm{x}}\in{\Omega},\\\
{{\mathcal{B}}_{\delta}}u=g_{\delta}({\bm{x}}),&\quad\forall\,{\bm{x}}\in{{\Omega}_{{\mathcal{I}}_{\delta}}},\end{cases}$
(12)

with $F$ and $g_{\delta}$ given and a nonlocal diffusion operator defined by

${{\mathcal{L}}_{\delta}}u{({\bm{x}})}:=\int_{{{\mathbb{R}}^{d}}}(u({\bm{y}})-u({\bm{x}})){\gamma_{\delta}}{({\bm{x}},{\bm{y}})}{d{\bm{y}}},\quad\quad\forall\,{\bm{x}}\in{{\mathbb{R}}^{d}},$
(13)

where the kernel function ${\gamma_{\delta}}{({\bm{x}},{\bm{y}})}$ is
symmetric, i.e.,

${\gamma_{\delta}}{({\bm{x}},{\bm{y}})}={\gamma_{\delta}}{({\bm{y}},{\bm{x}})},\quad\forall{\bm{x}},{\bm{y}}\in{{\mathbb{R}}^{d}},$

non-negative, and with a compact support in ${{\bm{y}}-{\bm{x}}}$ in
$B_{\delta}({\bf 0})$, the $\delta$-ball of the origin. that is,

${\gamma_{\delta}}{({\bm{x}},{\bm{y}})}=0\text{ whenever
}|{{\bm{y}}-{\bm{x}}}|>{\delta}.$

Again, the integral in (13) (and the one in (19), which will be introduced
shortly) should be understood in the sense of principal value throughout the
paper. Moreover, due to the compact support of ${\gamma_{\delta}}$, for
${\bm{x}}\in{\Omega}$, we can rewrite (13) as

${{\mathcal{L}}_{\delta}}u({\bm{x}}):=\int_{{{\widehat{{\Omega}}}_{\delta}}}(u({\bm{y}})-u({\bm{x}})){\gamma_{\delta}}{({\bm{x}},{\bm{y}})}{d{\bm{y}}},\quad\quad\forall\,{\bm{x}}\in{\Omega}.$
(14)

Analogous to (4), we have the choice of volume constraints

${{\mathcal{B}}_{\delta}}u({\bm{x}})=\begin{cases}u({\bm{x}})&{\bm{x}}\in{{\Omega}^{D}_{{\mathcal{I}}_{\delta}}}\;\mbox{(Dirichlet
VC)},\\\
{{\mathcal{N}}_{\delta}}u({\bm{x}}),&{\bm{x}}\in{{\Omega}^{N}_{{\mathcal{I}}_{\delta}}}\;\mbox{(Neumann
VC)},\end{cases}$ (15)

where ${{\Omega}^{D}_{{\mathcal{I}}_{\delta}}}$ and
${{\Omega}^{N}_{{\mathcal{I}}_{\delta}}}$ are two disjoint subdomains of
${{\Omega}_{{\mathcal{I}}_{\delta}}}$ such that
$\overline{{{\Omega}^{D}_{{\mathcal{I}}_{\delta}}}\cup{{\Omega}^{N}_{{\mathcal{I}}_{\delta}}}}=\overline{{{\Omega}_{{\mathcal{I}}_{\delta}}}}$.
The operator ${{\mathcal{N}}_{\delta}}$ can be defined, for example,
analogously to (10) by

${{\mathcal{N}}_{\delta}}u({\bm{x}}):=-\int_{\Omega}(u({\bm{y}})-u({\bm{x}})){\gamma_{\delta}}{({\bm{x}},{\bm{y}})}{d{\bm{y}}},\quad\forall\,{\bm{x}}\in{{\Omega}_{{\mathcal{I}}_{\delta}}}.$
(16)

Note that generally, ${{\mathcal{N}}_{\delta}}u$ only needs to be defined and
applied on ${{\Omega}^{N}_{{\mathcal{I}}_{\delta}}}$. Its definition on
${{\Omega}_{{\mathcal{I}}_{\delta}}}$ is given for the convenience of notation
only.

Rather than considering kernels in the general form, we focus on those
translation-invariant (i.e., functions of ${{\bm{y}}-{\bm{x}}}$) and radial
symmetric (i.e., depending only on ${|{{\bm{y}}-{\bm{x}}}|}$). In particular,
we consider a more specific re-scaled form given by

${\gamma_{\delta}}{({\bm{x}},{\bm{y}})}=C_{d}({\delta})\gamma\biggl{(}\displaystyle\frac{|{{\bm{y}}-{\bm{x}}}|}{{\delta}}\biggr{)}$
(17)

for a kernel $\gamma({|\cdot|})$ that satisfies

$\left\\{\begin{array}[]{l}\gamma(s)\geq 0,\text{ and non-increasing, }\forall
s\in[0,1),\;\;\gamma(s)=0,\forall s\geq 1,\\\
\widehat{\gamma}(|{\bm{z}}|)=|{\bm{z}}|^{2}\gamma(|{\bm{z}}|)\in
L^{1}_{\text{loc}}({{\mathbb{R}}^{d}}).\end{array}\right.$ (18)

For such kernels, we see that ${{\mathcal{L}}_{\delta}}$ can also be written
as

${{\mathcal{L}}_{\delta}}u{({\bm{x}})}:=C_{d}({\delta})\int_{{{\mathbb{R}}^{d}}}(u({\bm{y}})-u({\bm{x}})){\gamma}\left(\frac{{|{{\bm{y}}-{\bm{x}}}|}}{\delta}\right){d{\bm{y}}},\quad\quad\forall\,{\bm{x}}\in{{\mathbb{R}}^{d}}.$
(19)

The scaling factor $C_{d}({\delta})$ is of special interest in this work and
is defined differently depending on the regimes that we study. We give two
examples below.

  * •

For $0<{\delta}\ll\text{diam\,}({\Omega})$, $C_{d}({\delta})$ is chosen to
satisfy

$\int_{{{\mathbb{R}}^{d}}}{|{{\bm{y}}-{\bm{x}}}|}^{2}{\gamma_{\delta}}{({\bm{x}},{\bm{y}})}d{\bm{y}}={{\delta}^{d+2}}C_{d}({\delta})\int_{{B_{1}({\bf
0})}}|\bm{\xi}|^{2}\gamma(|\bm{\xi}|){d}\bm{\xi}=2d.$ (20)

so that as ${\delta}\to 0$, the local limit of ${{\mathcal{L}}_{\delta}}$
becomes the standard Laplacian ${{\mathcal{L}}_{0}}=\Delta$ [30, 28, 54, 55].

  * •

For ${\delta}\gg\text{diam\,}({\Omega})$, the factor and kernels are chosen to
satisfy

$C_{d}({\delta})=\frac{C_{d,s}}{{\delta}^{d+2s}}\quad\text{for}\quad{\gamma}(t)=t^{-{d}-2s}\mathbbm{1}_{(0,1)},$
(21)

so that
$C_{d}({\delta}){\gamma}(t/{\delta})=C_{d,s}t^{-d-2s}\mathbbm{1}_{(0,{\delta})}$,
that is, ${\gamma_{\delta}}{({\bm{x}},{\bm{y}})}$ is a truncation of the
fractional kernel ${{\gamma}_{s}}$ defined in (7)

${\gamma_{\delta}}{({\bm{x}},{\bm{y}})}=\begin{cases}{{\gamma}_{s}}{(|{{\bm{y}}-{\bm{x}}}|)}=\displaystyle\frac{{C_{{d},s}}}{{|{{\bm{y}}-{\bm{x}}}|}^{{d}+2s}},&\quad\forall\>{{\bm{y}}-{\bm{x}}}\in
B_{\delta}({\bf 0}),\\\ 0,&\quad\text{otherwise.}\end{cases}$ (22)

Consequently, we have
${{\mathcal{L}}_{\delta}}\to{{\mathcal{L}}_{\infty}^{s}}=-(-\Delta)^{s}$ as
${\delta}\to\infty$ [22, 12, 75].

######  Remark 2.4.

Although ${{\mathcal{L}}_{\delta}}u({\bm{x}})$ is defined for any
${\bm{x}}\in{{\mathbb{R}}^{d}}$, independent of any spatial domain, for
problem (12), we only need ${{\mathcal{L}}_{\delta}}u{({\bm{x}})}$ for
${\bm{x}}\in{\Omega}$. Then, with a compactly supported ${\gamma_{\delta}}$,
the integral in (19) can be equivalently defined over
${{\widehat{{\Omega}}}_{\delta}}$ instead, i.e.,

${{\mathcal{L}}_{\delta}}u{({\bm{x}})}:=\int_{{{\widehat{{\Omega}}}_{\delta}}}(u({\bm{y}})-u({\bm{x}})){\gamma_{\delta}}{({\bm{x}},{\bm{y}})}{d{\bm{y}}},\quad\quad\forall\,{\bm{x}}\in{\Omega}.$
(23)

######  Remark 2.5.

The operator ${{\mathcal{L}}_{\delta}}$ defined by (19), with a kernel scaled
by (20) or (22), is a nonlocal analog of the partial differential operator
${{\mathcal{L}}_{0}}=\Delta$ or the fractional operator
${{\mathcal{L}}_{\infty}^{s}}=-(-\Delta)^{s}$. Likewise, the nonlocal model
(12) is a nonlocal analog of the local diffusion model (3) and the fractional
diffusion model (8). Nonlocal diffusion problems have also been extensively
studied in recent years, we refer to the literature review given in [30] and
[20]. In addition, one can find additional references in a few more recent
works [23, 45, 60, 78].

######  Remark 2.6.

We note that, different from [20, 31], the operators
${{\mathcal{L}}_{\delta}}$ and ${{\mathcal{N}}_{\delta}}$, involving the
respective domains of integration in their definitions, are adopted as in [28,
33] to be consistent with the fractional version, see e.g. [26, 43], given by
(10). We also consider their variants, as well as the variants of the
fractional operators later in the paper.

######  Remark 2.7.

Concerning the operator ${{\mathcal{B}}_{\delta}}$ in (15), one may consider
more general forms. For example, with
${{\Omega}_{{\mathcal{I}}_{\delta}}}={{\Omega}^{D}_{{\mathcal{I}}_{\delta}}}$,
we may have

${{\mathcal{B}}_{\delta}}u({\bm{x}})={{\mathcal{N}}_{\delta}}u({\bm{x}})-{{{\mathcal{D}}_{\delta}}}u({\bm{x}}),\;\forall{\bm{x}}\in{{\Omega}_{{\mathcal{I}}_{\delta}}}$
(24)

where the linear operator ${{\mathcal{D}}_{\delta}}$ is given by

${{{\mathcal{D}}_{\delta}}}u({\bm{x}})=\int_{{{\Omega}_{{\mathcal{I}}_{\delta}}}}u({\bm{y}})\sigma_{\delta}({\bm{x}},{\bm{y}})d{\bm{y}},\;\forall{\bm{x}}\in{{\Omega}_{{\mathcal{I}}_{\delta}}}={{\Omega}^{D}_{{\mathcal{I}}_{\delta}}}$
(25)

where the $\sigma_{\delta}({\bm{x}},{\bm{y}})$ can be a bounded and symmetric
kernel such that as $\delta\to 0$,

$\int_{{{\Omega}_{{\mathcal{I}}_{\delta}}}}{{{\mathcal{D}}_{\delta}}}u({\bm{x}})v({\bm{x}})d{\bm{x}}\to\int_{\partial{\Omega}}u({\bm{x}})v({\bm{x}})d{\bm{x}},$
(26)

for any pair of smoothly defined functions $u$ and $v$ on ${\Omega}$. In this
case, we get a problem analogous to the local Robin boundary value problem.

######  Remark 2.8.

While we focus on nonlocal diffusion operators defined by (19) with a
symmetric kernel for illustration, we note that more general nonlocal
diffusion-convection operators with a nonsymmetric kernel have also been
studied, see, for instance, [32, 48, 70, 71, 21].

Note that for the pure Neumann case, i.e.,
${{\Omega}^{N}_{{\mathcal{I}}_{\delta}}}={{{\Omega}_{{\mathcal{I}}_{\delta}}}}$,
we impose a constraint on the solution $u$ to have mean $0$ over
${{\widehat{{\Omega}}}_{\delta}}$ and the equations in (12) are modified to

$\begin{cases}-{{\mathcal{L}}_{\delta}}u=F({\bm{x}},u({\bm{x}}))-\lambda_{N},&\quad\forall\,{\bm{x}}\in{\Omega},\\\
{{\mathcal{N}}_{\delta}}u={g_{N}^{\delta}}({\bm{x}})-\lambda_{N},&\quad\forall\,{\bm{x}}\in{{\Omega}_{{\mathcal{I}}_{\delta}}},\end{cases}$
(27)

where $\lambda_{N}$ is a constant Lagrange multiplier satisfying

$\displaystyle\int_{\Omega}F({\bm{x}},u({\bm{x}}))\,{d{\bm{x}}}+\int_{{\Omega}_{{\mathcal{I}}_{\delta}}}{g_{N}^{\delta}}({\bm{x}})=\lambda_{N}|{{\widehat{{\Omega}}}_{\delta}}|,$
(28)

where $|{{\widehat{{\Omega}}}_{\delta}}|$ denotes the nonzero measure of
${{\widehat{{\Omega}}}_{\delta}}$.

###  2.4 Regional fractional and nonlocal Laplacians

An operator related to the fractional Laplacian, but defined with respect to
the domain ${\Omega}$, is the regional fractional Laplacian given by [7, 14]

$(-\Delta)_{\Omega}^{s}u({{\bm{x}}})=\int_{{\Omega}}(u({{\bm{x}}})-u({{\bm{y}}})){{\gamma}_{s}}{({\bm{x}},{\bm{y}})}{d{\bm{y}}}{,}\quad\forall\,{\bm{x}}\in{\Omega}.$
(29)

It can be used to define a problem given by

$(-\Delta)_{\Omega}^{s}u({{\bm{x}}})=F({\bm{x}},u({\bm{x}})),\quad\quad\forall\,{\bm{x}}\in{\Omega},$
(30)

which is another approach to generalize the local Poisson Neumann problem to
the nonlocal fractional case [26, Section 7]. Note that in comparison with the
fractional Laplacian ${{\mathcal{L}}_{\infty}^{s}}$ defined in (6), the only
difference is that the regional operator in (29) involves an integral over
${\Omega}$ instead of ${{\mathbb{R}}^{d}}$.

Similarly, a nonlocal problem on ${\Omega}$ related to the regional fractional
Laplacian is

$-{{\mathcal{L}}_{{\delta},\Omega}}u({\bm{x}})=F({\bm{x}},u({\bm{x}})),\quad\quad\forall\,{\bm{x}}\in{\Omega},$
(31)

where the regional nonlocal diffusion operator
${{\mathcal{L}}_{{\delta},\Omega}}$ is given by

${{\mathcal{L}}_{{\delta},\Omega}}u({\bm{x}})=\int_{{\Omega}}(u({{\bm{y}}})-u({{\bm{x}}})){\gamma_{\delta}}{({\bm{x}},{\bm{y}})}{d{\bm{y}}}{,}\quad\quad\forall\,{\bm{x}}\in{\Omega}.$
(32)

Then (30) is a special case of (31) by choosing special fractional kernels as
in (21) and setting $\delta=\infty$. Hence, for kernels of the form (21), we
may see ${{\mathcal{L}}_{{\delta},\Omega}}$ as a “regional” nonlocal Laplacian
with respect to the domain ${\Omega}$.

Indeed, let ${\Omega}={\Omega}_{1}\cup{\Omega}_{2}$ with
${\Omega}_{1}\cap{\Omega}_{2}=\emptyset$, the problem (31), by a simple change
of notations, can be equivalently written as the nonlocal problem:

$\left\\{\begin{array}[]{ll}-{{\mathcal{L}}_{{\delta},\Omega}}u({\bm{x}})=F({\bm{x}},u({\bm{x}}))&\quad\forall\,{\bm{x}}\in{\Omega}_{1},\\\
{{\mathcal{N}}_{{\delta},{\Omega}}}u({\bm{x}})=G({\bm{x}},u({\bm{x}}))&\quad\forall\,{\bm{x}}\in{\Omega}_{2}.\end{array}\right.$
(33)

Here, a nonlocal operator ${{\mathcal{N}}_{{\delta},{\Omega}}}$ is introduced,
which is defined by

${{\mathcal{N}}_{{\delta},{\Omega}}}u({\bm{x}}):=-{{\mathcal{L}}_{{\delta},\Omega}}u({\bm{x}})=\int_{{\Omega}}(u({\bm{x}})-u({\bm{y}})){\gamma_{\delta}}{({\bm{x}},{\bm{y}})}{d{\bm{y}}},\quad\forall{\bm{x}}\in{\Omega}_{2},$
(34)

and $G({\bm{x}},u({\bm{x}}))=F({\bm{x}},u({\bm{x}}))$ on ${\Omega}_{2}$.
${{\mathcal{N}}_{{\delta},{\Omega}}}$ and $G({\bm{x}},u({\bm{x}}))$ only carry
symbolic meanings in this case and the equivalence of (31) and (33) holds
independently of how ${\Omega}$ is decomposed into ${\Omega}_{1}$ and
${\Omega}_{2}$. For example, a special choice of the domain ${\Omega}_{2}$,
for $\delta$ sufficiently small, is the interior ${\delta}$-layer of
${\Omega}$ given by

$\displaystyle{{\Omega}_{{\mathcal{I}}^{0}_{\delta}}}=\\{{\bm{x}}\in{\Omega}\>\mid\>\text{dist}({\bm{x}},{\partial{\Omega}})<{\delta}\,\\}.$
(35)

######  Remark 2.9.

Note that the definition (34) of ${{\mathcal{N}}_{{\delta},{\Omega}}}$ has
been adopted in discussions in [20, 31], except for the use of different
notation for the geometric domains. It can be seen that
${{\mathcal{N}}_{{\delta},{\Omega}}}$ adopts the same definition as
$-{{\mathcal{L}}_{{\delta},\Omega}}$ except that while
$-{{\mathcal{L}}_{{\delta},\Omega}}u({\bm{x}})$ is applied on ${\Omega}_{1}$,
${{\mathcal{N}}_{{\delta},{\Omega}}}u({\bm{x}})$ inherits the definition on
${\Omega}_{2}$.

######  Remark 2.10.

We note that the reformulation (33) of (31) resembles more closely, in
appearance, the conventional way of presenting the problem as an equation
coupled with a boundary condition (or say volume constraint). Yet, the
equivalence of the formulations also highlights the fact that for nonlocal
problems, the notion of boundary conditions can be superfluous. A more general
way to understand the formulation is that a nonlocal model is largely
described by the law of nonlocal interactions that could be altered due to the
presence of a boundary.

######  Remark 2.11.

We also note that the nonlocal diffusion operator ${{\mathcal{L}}_{\delta}}$
reformulated in (23), with the integral being over
${{\widehat{{\Omega}}}_{\delta}}$ instead of ${{\mathbb{R}}^{d}}$ for
compactly supported kernels of an interaction range $\delta$, may have the
appearance of being associated with a “regional” nonlocal diffusion operator
defined on the combined domain ${{\widehat{{\Omega}}}_{\delta}}$. However,
this association can only be meaningful in the case where the equation (23),
rather than being confined to ${\Omega}$, is extended to all
${\bm{x}}\in{{\widehat{{\Omega}}}_{\delta}}$.

Let us examine some limiting cases formally. First, with the kernel
${\gamma_{\delta}}{({\bm{x}},{\bm{y}})}$ properly scaled with respect to the
horizon parameter $\delta$ as in (20), we see that (31), thus (33), is
expected to converge as ${\delta}\to 0$ to a Neumann problem of the local
diffusion operator with the corresponding boundary condition determined by the
limiting behavior of $F$ on the interior ${\delta}$-layer
${{\Omega}_{{\mathcal{I}}^{0}_{\delta}}}$ defined in (35).

Likewise, with ${\gamma_{\delta}}{({\bm{x}},{\bm{y}})}$ chosen as in (22),
then (33) is expected to recover exactly the regional fractional Laplacian
Neumann problem (30) if ${\delta}>\text{diam}({\Omega})$. We thus see the
regional nonlocal Laplacian again provides a bridge between the local
Laplacian and the regional fractional Laplacian.

We note that spectral decomposition is another approach to define fractional
differential operators and nonlocal integral operators. Generically, such
spectrally represented operators might not yield operators with a finite range
of interactions spatially. In many cases, the resulting nonlocal kernels for
their spatial integral formulations might not have simple analytical
representations in real space. We thus do not consider further in this
direction, interested readers can find additional discussions in [1, 8, 13,
43, 59, 67, 62, 5, 19].

###  2.5 Nonlocal flux and Nonlocal Green’s identities

In [31, 20], the operator ${{\mathcal{N}}_{{\delta},{\Omega}}}$ is also used
to provide a nonlocal Green’s formula that shares more symbolic resemblance to
the classical Green’s formula for the local Laplacian and for defining a
nonlocal flux. We provide a more rigorous statement and derivation here for
clarity.

######  Proposition 2.12.

Given the nonlocal operators defined by a rescaled kernel (17) with the
conditions (18) and (20) satisfied, for functions $u$ and $v$ in
$C^{2}(\overline{{\Omega}\cup{\Omega}^{\prime}})$, we have the nonlocal
Green’s first identity given by

$\displaystyle\frac{1}{2}\int_{({\Omega}\cup{\Omega}^{\prime})^{2}}(v({\bm{y}})-v({\bm{x}}))(u({\bm{y}})-u({\bm{x}})){\gamma_{\delta}}{({\bm{x}},{\bm{y}})}{d{\bm{y}}d{\bm{x}}}=-\int_{{\Omega}\cup{\Omega}^{\prime}}v({\bm{x}}){\mathcal{L}}_{{\delta},{\Omega}\cup{\Omega}^{\prime}}u({\bm{x}}){d{\bm{x}}}$
$\displaystyle\quad=-\int_{{\Omega}}v({\bm{x}}){\mathcal{L}}_{{\delta},{\Omega}\cup{\Omega}^{\prime}}u({\bm{x}}){d{\bm{x}}}+\int_{{\Omega}^{\prime}}v({\bm{x}}){\mathcal{N}}_{{\delta},{\Omega}\cup{\Omega}^{\prime}}u({\bm{x}}){d{\bm{x}}}.$
(36)

This also readily implies the also the nonlocal Green’s second identity

$\displaystyle\int_{{\Omega}}v({\bm{x}}){\mathcal{L}}_{{\delta},{\Omega}\cup{\Omega}^{\prime}}u({\bm{x}}){d{\bm{x}}}-\int_{{\Omega}}u({\bm{x}}){\mathcal{L}}_{{\delta},{\Omega}\cup{\Omega}^{\prime}}v({\bm{x}}){d{\bm{x}}}$
$\displaystyle\quad=-\int_{{\Omega}^{\prime}}u({\bm{x}}){\mathcal{N}}_{{\delta},{\Omega}\cup{\Omega}^{\prime}}v({\bm{x}}){d{\bm{x}}}+\int_{{\Omega}^{\prime}}v({\bm{x}}){\mathcal{N}}_{{\delta},{\Omega}\cup{\Omega}^{\prime}}u({\bm{x}}){d{\bm{x}}}.$
(37)

###### Proof.

To prove the Proposition 2.12, the key is to show that the integrals are well-
defined so that the classical Fubini theorem can be applied to get (2.12). For
$u\in C^{2}(\overline{{\Omega}\cup{\Omega}^{\prime}})$, we now check that
${\mathcal{L}}_{{\delta},{\Omega}\cup{\Omega}^{\prime}}u\in
L^{1}({\Omega}\cup{\Omega}^{\prime})$, which is a sufficient condition to
achieve this. To simplify the notation, we use ${\Omega}$ to replace
${\Omega}\cup{\Omega}^{\prime}$ so that

$\displaystyle{\mathcal{L}}_{{\delta},{\Omega}}u({\bm{x}})=\int_{\Omega}(u({\bm{y}})-u({\bm{x}})){\gamma_{\delta}}{({\bm{x}},{\bm{y}})}d{\bm{y}}.$

Then it suffices to show that ${\mathcal{L}}_{{\delta},{\Omega}}u\in
L^{1}({\Omega})$ for all $u\in C^{2}(\overline{{\Omega}})$.

By the regularity of $u$, we have for ${\bm{x}}\in{\Omega}$,

$\begin{split}{\mathcal{L}}_{{\delta},{\Omega}}u({\bm{x}})&=\int_{{\Omega}\backslash
B_{{\rm{dist}}({\bm{x}},\partial{\Omega})}({\bm{x}})}(u({\bm{y}})-u({\bm{x}})){\gamma_{\delta}}{({\bm{x}},{\bm{y}})}d{\bm{y}}\\\
&\qquad+\int_{B_{{\rm{dist}}({\bm{x}},\partial{\Omega})}({\bm{x}})}(u({\bm{y}})-u({\bm{x}})){\gamma_{\delta}}(|{\bm{y}}-{\bm{x}}|)d{\bm{y}}\\\
&\leq C\int_{{\Omega}\backslash
B_{{\rm{dist}}({\bm{x}},\partial{\Omega})}({\bm{x}})}|{\bm{y}}-{\bm{x}}|{\gamma_{\delta}}(|{\bm{y}}-{\bm{x}}|)d{\bm{y}}\\\
&\qquad+C\int_{B_{{\rm{dist}}({\bm{x}},\partial{\Omega})}({\bm{x}})}|{\bm{y}}-{\bm{x}}|^{2}{\gamma_{\delta}}(|{\bm{y}}-{\bm{x}}|)d{\bm{y}}\\\
&\leq
C\left(\int_{{\rm{dist}}({\bm{x}},\partial{\Omega})<|{\bm{z}}|<{\delta}}|{\bm{z}}|{\gamma_{\delta}}(|{\bm{z}}|)d{\bm{z}}+1\right)\end{split}$

where the constant $C$ depends on $\|u\|_{C^{2}(\overline{{\Omega}})}$. Thus,
the task remains is to show that

$w({\bm{x}}):=\int_{{\rm{dist}}({\bm{x}},\partial{\Omega})<|{\bm{z}}|<{\delta}}|{\bm{z}}|{\gamma_{\delta}}(|{\bm{z}}|)d{\bm{z}}\in
L^{1}({\Omega})\,.$

Let ${\Omega}^{\epsilon}$ denote an inner layer of width $\epsilon>0$
surrounding $\partial{\Omega}$ where $\epsilon>0$ is a small fixed number.
Since it is obvious that $w\in L^{1}({\Omega}\backslash{\Omega}^{\epsilon})$,
we only need to show $w\in L^{1}({\Omega}^{\epsilon})$. Define for
$t\in(0,\epsilon)$,

$\partial{\Omega}_{t}:=\\{{\bm{x}}\in{\Omega}~{}|~{}{\rm{dist}}({\bm{x}},\partial{\Omega})=t\\}.$

Then, there exists a general constant $C>0$,

$\begin{split}\int_{{\Omega}^{\epsilon}}\int_{{\rm{dist}}({\bm{x}},\partial{\Omega})<|{\bm{z}}|<{\delta}}|{\bm{z}}|{\gamma_{\delta}}(|{\bm{z}}|)d{\bm{z}}&\leq
C\int_{0}^{\epsilon}\int_{\partial{\Omega}_{t}}\int_{t<r<{\delta}}r^{d}{\gamma_{\delta}}(r)drd{\bm{x}}dt\\\
&\leq C\int_{0}^{\epsilon}\int_{t<r<{\delta}}r^{d}{\gamma_{\delta}}(r)drdt\\\
&=C\int_{0}^{\delta}\int_{0}^{\min(r,\epsilon)}r^{d}{\gamma_{\delta}}(r)dtdr\\\
&\leq C\int_{0}^{\delta}r^{d+1}{\gamma_{\delta}}(r)dr<\infty.\end{split}$

Thus, we get the desired result that ${\mathcal{L}}_{{\delta},{\Omega}}u\in
L^{1}({\Omega})$. ∎

######  Remark 2.13.

Note that by density argument, one can show that the above identities hold in
the closure of $C^{2}(\overline{{\Omega}\cup{\Omega}^{\prime}})$ with respect
to suitable norms.

######  Remark 2.14.

In [31] and [28], the identity (2.12) is stated as for
${\Omega}={\Omega}_{1}\cup{\Omega}_{2}$,

$\displaystyle\int_{{\Omega}}v({\bm{x}}){{\mathcal{L}}_{{\delta},\Omega}}u({\bm{x}}){d{\bm{x}}}=\int_{{\Omega}}u({\bm{x}}){{\mathcal{L}}_{{\delta},\Omega}}v({\bm{x}}){d{\bm{x}}}.$
(38)

This relation, as a consequence of Fubini’s theorem, can be seen as a nonlocal
version of integration by parts.

It is worth pointing out that a similar role as
${{\mathcal{N}}_{{\delta},{\Omega}}}$ can be played by the operator
${{\mathcal{N}}_{\delta}}$ defined earlier in (16). In fact, corresponding to
the function $u$ defined on a union ${\Omega}\cup{\Omega}^{\prime}$ of domains
${\Omega}$ and ${\Omega}^{\prime}$, let ${{\mathcal{N}}_{\delta}}u({\bm{x}})$
be defined on ${\Omega}^{\prime}$ in the same way as (16), the so-called
nonlocal flux of $u$ between the domain ${\Omega}$ and another domain
${\Omega}^{\prime}$, which is induced from ${{\mathcal{N}}_{\delta}}$, can be
specified as

$\displaystyle\mathcal{F}({\Omega},{\Omega}^{\prime})=\int_{{\Omega}^{\prime}}\int_{{\Omega}}(u({\bm{x}})-u({\bm{y}})){\gamma_{\delta}}{({\bm{x}},{\bm{y}})}\,d{\bm{y}}\,d{\bm{x}}=\int_{{\Omega}^{\prime}}{{\mathcal{N}}_{\delta}}u({\bm{x}})\,d{\bm{x}}.$

Then for a symmetric kernel ${\gamma_{\delta}}{({\bm{x}},{\bm{y}})}$, i.e.,
${\gamma_{\delta}}{({\bm{x}},{\bm{y}})}={\gamma_{\delta}}{({\bm{y}},{\bm{x}})}$,
we have the following properties.

  * •

Anti-symmetry:

$\displaystyle\mathcal{F}({\Omega},{\Omega}^{\prime})=-\mathcal{F}({\Omega}^{\prime},{\Omega})=-\int_{{\Omega}}\int_{{\Omega}^{\prime}}(u({\bm{y}})-u({\bm{x}})){\gamma_{\delta}}{({\bm{y}},{\bm{x}})}\,{d{\bm{x}}}\,{d{\bm{y}}},\quad$

holds formally (assuming that Fubini’s theorem is applicable).

  * •

Additivity:

$\displaystyle\mathcal{F}({\Omega}_{1},{\Omega}^{\prime})+\mathcal{F}({\Omega}_{2},{\Omega}^{\prime})=\mathcal{F}({\Omega},{\Omega}^{\prime})\quad\text{for
}\;{\Omega}={\Omega}_{1}\cup{\Omega}_{2},\;{\Omega}_{1}\cap{\Omega}_{2}=\emptyset.$

######  Remark 2.15.

With $\mathcal{F}({\Omega}_{1},{\Omega}_{1})=0$, we see that

$\displaystyle\mathcal{F}({\Omega}_{1},{\Omega}_{2})$
$\displaystyle=\mathcal{F}({\Omega}_{1},{\Omega}_{2})+\mathcal{F}({\Omega}_{2},{\Omega}_{2})=\mathcal{F}({\Omega}_{1}\cup{\Omega}_{2},{\Omega}_{2})$
$\displaystyle=\int_{{\Omega}_{2}}\int_{{\Omega}}(u({\bm{x}})-u({\bm{y}})){\gamma_{\delta}}{({\bm{x}},{\bm{y}})}\,d{\bm{y}}\,d{\bm{x}}=\int_{{\Omega}_{2}}{{\mathcal{N}}_{{\delta},{\Omega}}}u({\bm{x}})\,d{\bm{x}}.$

This also implies that ${{\mathcal{N}}_{\delta}}$ induces the same flux
functional as ${{\mathcal{N}}_{{\delta},{\Omega}}}$ for any symmetric kernel.

Next, let us state the nonlocal Green’s first identity given by

$\displaystyle\frac{1}{2}\int_{({\Omega}\cup{\Omega}^{\prime})^{2}\setminus{\Omega}^{\prime
2}}(v({\bm{y}})-v({\bm{x}}))(u({\bm{y}})-u({\bm{x}})){\gamma_{\delta}}{({\bm{x}},{\bm{y}})}{d{\bm{y}}d{\bm{x}}}$
$\displaystyle\quad=-\int_{{\Omega}}v({\bm{x}}){\mathcal{L}}_{{\delta},{\Omega}\cup{\Omega}^{\prime}}u({\bm{x}}){d{\bm{x}}}+\int_{{\Omega}^{\prime}}v({\bm{x}}){{\mathcal{N}}_{\delta}}u({\bm{x}}){d{\bm{x}}}.$
(39)

In particular, if ${\Omega}^{\prime}={{\Omega}_{{\mathcal{I}}_{\delta}}}$,
then with
${\widetilde{{\Omega}}_{\delta}}={{{\widehat{{\Omega}}}_{\delta}}^{2}\setminus{\Omega}^{2}_{{\mathcal{I}}_{\delta}}}$,
we have

$\displaystyle\frac{1}{2}\int_{{\widetilde{{\Omega}}_{\delta}}}(v({\bm{y}})-v({\bm{x}}))(u({\bm{y}})-u({\bm{x}})){\gamma_{\delta}}{({\bm{x}},{\bm{y}})}{d{\bm{y}}d{\bm{x}}}$
$\displaystyle\quad=-\int_{{\Omega}}v({\bm{x}}){{\mathcal{L}}_{\delta}}u({\bm{x}}){d{\bm{x}}}+\int_{{{\Omega}_{{\mathcal{I}}_{\delta}}}}v({\bm{x}}){{\mathcal{N}}_{\delta}}u({\bm{x}}){d{\bm{x}}}.$
(40)

Note that the identity (2.5) is different from (2.12) as given in [31, 20],
but it is a more natural nonlocal analog of the (generalized) local Green’s
first identity

$\displaystyle\int_{\Omega}\nabla v({\bm{x}})\cdot\nabla
u({\bm{x}}){d{\bm{x}}}=-\int_{\Omega}v({\bm{x}})\Delta
u({\bm{x}}){d{\bm{x}}}+\int_{\partial{\Omega}}v({\bm{x}})\frac{\partial
u}{\partial\mathbf{n}}({\bm{x}}){d{\bm{x}}},$ (41)

as well as the fractional version given by

$\displaystyle\frac{1}{2}\int_{{\widetilde{{\Omega}}_{\infty}}}(v({\bm{y}})-v({\bm{x}}))(u({\bm{y}})-u({\bm{x}})){{\gamma}_{s}}{(|{{\bm{y}}-{\bm{x}}}|)}{d{\bm{y}}d{\bm{x}}}$
$\displaystyle\quad=-\int_{{\Omega}}v({\bm{x}}){{\mathcal{L}}_{\infty}^{s}}u({\bm{x}}){d{\bm{x}}}+\int_{{{\Omega}_{c}}}v({\bm{x}}){{\mathcal{N}}_{\infty}^{s}}u({\bm{x}}){d{\bm{x}}},$
(42)

where
${\widetilde{{\Omega}}_{\infty}}=({{\mathbb{R}}^{d}})^{2}\setminus{\Omega}_{c}^{2}$.

For smooth function $u$, by simple Taylor expansion, we have
${{\mathcal{L}}_{\delta}}u({\bm{x}})\to\Delta u({\bm{x}})$ in ${\Omega}$ if we
choose the kernels according to (20). In this case, from the first Green’s
identity (2.5), we may formally let $\delta\to 0$ to get that

$\displaystyle\int_{{{\Omega}_{{\mathcal{I}}_{\delta}}}}v({\bm{x}}){{\mathcal{N}}_{\delta}}u({\bm{x}}){d{\bm{x}}}\to\int_{\partial{\Omega}}v({\bm{x}})\frac{\partial
u}{\partial\mathbf{n}}d{\bm{x}}.$ (43)

This offers some justification to the statement that the local version (41) of
(2.5) can be seen as the ${\delta}\to 0$ limit of (2.5). In this case, from
the above formal argument, we expect that the Neumann operator
${{\mathcal{N}}_{\delta}}$ converges to the local one in the distribution
sense. Meanwhile, the fractional version can be seen as the
${\delta}\to\infty$ limit if the kernel is given according to (22).
Furthermore, as $\delta\to 0$, formally we may also note that

$\displaystyle\int_{{{\Omega}_{{\mathcal{I}}_{\delta}}}}v({\bm{x}}){\mathcal{N}}_{\delta,{{\widehat{{\Omega}}}_{\delta}}}u({\bm{x}}){d{\bm{x}}}\to\int_{\partial{\Omega}}v({\bm{x}})\frac{\partial
u}{\partial\mathbf{n}}d{\bm{x}}.$ (44)

Indeed, the difference between the left-hand side integrals of (43) and (44)
satisfies

$\displaystyle\int_{{{\Omega}_{{\mathcal{I}}_{\delta}}}}v({\bm{x}})\int_{{{\Omega}_{{\mathcal{I}}_{\delta}}}}(u({\bm{x}})-u({\bm{y}}))\gamma_{\delta}(|{\bm{x}}-{\bm{y}}|){d{\bm{y}}}{d{\bm{x}}}$
$\displaystyle\quad=\frac{1}{2}\int_{{{\Omega}_{{\mathcal{I}}_{\delta}}}}\int_{{{\Omega}_{{\mathcal{I}}_{\delta}}}}(u({\bm{x}})-u({\bm{y}}))(v({\bm{x}})-v({\bm{y}}))\gamma_{\delta}(|{\bm{x}}-{\bm{y}}|){d{\bm{y}}}{d{\bm{x}}}=O(\delta)$

for smooth functions $u$ and $v$ defined on the whole space. This observation
shows the formal consistency using the different Neumann operators in the
local limit. However, We note that consistency to the fractional limit, unlike
the case of ${{\mathcal{N}}_{\delta}}$ is not shared by the nonlocal analog
presented in [31, 20] based on the operator
${\mathcal{N}}_{\delta,{{\widehat{{\Omega}}}_{\delta}}}$.

Likewise, the nonlocal analog of

$\int_{\Omega}\left[v({\bm{x}})\Delta u({\bm{x}})-u({\bm{x}})\Delta
v({\bm{x}})\right]{d{\bm{x}}}=\int_{\partial{\Omega}}\left[v({\bm{x}})\frac{\partial
u}{\partial\mathbf{n}}({\bm{x}})-u({\bm{x}})\frac{\partial
v}{\partial\mathbf{n}}({\bm{x}})\right]{d{\bm{x}}}$

is the nonlocal Green’s second identity given by

$\displaystyle\int_{{\Omega}}[v({\bm{x}}){{\mathcal{L}}_{\delta}}u({\bm{x}})-u({\bm{x}}){{\mathcal{L}}_{\delta}}v({\bm{x}})]{d{\bm{x}}}=\int_{{{\Omega}_{{\mathcal{I}}_{\delta}}}}[v({\bm{x}}){{\mathcal{N}}_{\delta}}u({\bm{x}})-u({\bm{x}}){{\mathcal{N}}_{\delta}}v({\bm{x}})]{d{\bm{x}}}.$
(45)

The factional analog can be easily obtained in the limit $\delta\to\infty$:

$\displaystyle\int_{{\Omega}}[v({\bm{x}}){{\mathcal{L}}_{\infty}^{s}}u({\bm{x}})-u({\bm{x}}){{\mathcal{L}}_{\infty}^{s}}v({\bm{x}})]{d{\bm{x}}}=\int_{{{\Omega}_{c}}}[v({\bm{x}}){{\mathcal{N}}_{\infty}^{s}}u({\bm{x}})-u({\bm{x}}){{\mathcal{N}}_{\infty}^{s}}v({\bm{x}})]{d{\bm{x}}}.$
(46)

######  Remark 2.16.

Rigorous proofs of the nonlocal Green’s identities and their connections to
the local and fractional analog in their full generality depend on the
characterization of the trace spaces, and we will leave the discussions to
future works, though we note some related works in the fractional case [43]
and recent works on the nonlocal trace theorems in [38].

##  3 Well-posed inhomogeneous nonlocal variational problems

We use linear problems where $F({\bm{x}},u)=f({\bm{x}})$ as illustrations for
well-posed nonlocal models with a horizon parameter ${\delta}$ and
inhomogeneous nonlocal constraints. We consider weak formulations
corresponding to (12) and make connections to a minimization principle. We pay
particular attention to the results that are independent of ${\delta}$ so that
the local and fractional cases can then become special cases, which enforces
the generality of the nonlocal models.

###  3.1 Function spaces and norms

Given a generic domain ${\Omega}$, a parameter $\delta>0$ and the extended
domain ${{\widehat{{\Omega}}}_{\delta}}$, let us first define an ‘energy’
space

${V_{{\delta}}({{\widehat{{\Omega}}}_{\delta}})}=\\{v({\bm{x}})\in
L^{2}({{\widehat{{\Omega}}}_{\delta}})\colon\|v\|_{{V_{{\delta}}({{\widehat{{\Omega}}}_{\delta}})}}<\infty\\},$
(47)

where $\|\cdot\|_{{V_{{\delta}}({{\widehat{{\Omega}}}_{\delta}})}}$ denotes an
‘energy’ norm induced by the inner product

${\langle\cdot,\cdot\rangle_{{V_{{\delta}}({{\widehat{{\Omega}}}_{\delta}})}}}=(\cdot,\cdot)_{{V_{{\delta}}({{\widehat{{\Omega}}}_{\delta}})}}+(\cdot,\cdot)_{{{\widehat{{\Omega}}}_{\delta}}}$

with $(\cdot,\cdot)_{{V_{{\delta}}({{\widehat{{\Omega}}}_{\delta}})}}$ being
defined as:

$\displaystyle(u,v)_{{V_{{\delta}}({{\widehat{{\Omega}}}_{\delta}})}}=\frac{1}{2}\int_{{\widetilde{{\Omega}}_{\delta}}}(v({\bm{y}})-v({\bm{x}}))(u({\bm{y}})-u({\bm{x}})){\gamma_{\delta}}{({\bm{x}},{\bm{y}})}\,{d{\bm{y}}d{\bm{x}}}\,,$
(48)

for any $u,v\in{V_{{\delta}}({{\widehat{{\Omega}}}_{\delta}})}$. Here
${\widetilde{{\Omega}}_{\delta}}={{{\widehat{{\Omega}}}_{\delta}}^{2}\setminus{\Omega}^{2}_{{\mathcal{I}}_{\delta}}}$.

######  Remark 3.1.

A different choice of the nonlocal space and its norm, defined over a domain
${\widehat{{\Omega}}}$, corresponds to the inner product

$\displaystyle{\langle
u,v\rangle}_{{V({\widehat{{\Omega}}})}}=(u,v)_{{V({\widehat{{\Omega}}})}}+(u,v)_{{\widehat{{\Omega}}}}$
(49)

where

$\displaystyle(u,v)_{{V({\widehat{{\Omega}}})}}=\frac{1}{2}\int_{{\widehat{{\Omega}}}}\int_{{\widehat{{\Omega}}}}(u({\bm{y}})-u({\bm{x}}))(v({\bm{y}})-v({\bm{x}})){\gamma_{\delta}}{({\bm{x}},{\bm{y}})}{d{\bm{y}}d{\bm{x}}}$
(50)

Such space and norm can be associated with the regional version of the
nonlocal and fractional versions of diffusion discussed in Section 2.4. When
${\widehat{{\Omega}}}$ is replaced by ${{\widehat{{\Omega}}}_{\delta}}$, the
main difference with the norm of
${V_{{\delta}}({{\widehat{{\Omega}}}_{\delta}})}$ induced from (48) is the
inclusion of additional nonlocal interactions in
${{\Omega}^{2}_{{\mathcal{I}}_{\delta}}}$ for the norm in
$V({{\widehat{{\Omega}}}_{\delta}})$. Note that this difference disappears for
functions constrained to have zero values on
${{\Omega}_{{\mathcal{I}}_{\delta}}}$.

The space ${V_{{\delta}}({{\widehat{{\Omega}}}_{\delta}})}$ is by definition a
subspace of $L^{2}({{\widehat{{\Omega}}}_{\delta}})$ and can be shown, for
suitably chosen kernels like those specified by (20) or (22), to be a Hilbert
space.

The constrained energy space is a closed subspace of
${V_{{\delta}}({{\widehat{{\Omega}}}_{\delta}})}$ defined by

${V_{c,{\delta}}({{\widehat{{\Omega}}}_{\delta}})}=\\{v({\bm{x}})\in{V_{{\delta}}({{\widehat{{\Omega}}}_{\delta}})}\colon
v({\bm{x}})=0\ \mbox{for
${\bm{x}}\in{{\Omega}^{D}_{{\mathcal{I}}_{\delta}}}$}\\}.$ (51)

Note that if ${{\Omega}^{D}_{{\mathcal{I}}_{\delta}}}=\emptyset$, we may pick
a nontrivial subset $K$ of ${{\widehat{{\Omega}}}_{\delta}}$ to define the
space by space ${V_{c,{\delta}}({{\widehat{{\Omega}}}_{\delta}})}$ by

${V_{c,{\delta}}({{\widehat{{\Omega}}}_{\delta}})}=\\{v({\bm{x}})\in{V_{{\delta}}({{\widehat{{\Omega}}}_{\delta}})}\>\colon\>\int_{K}v({\bm{x}})d{\bm{x}}=0\;\\}.$
(52)

Some natural choices include $K={\Omega}$ and
$K={{\widehat{{\Omega}}}_{\delta}}$. The dual space of
${V_{c,{\delta}}({{\widehat{{\Omega}}}_{\delta}})}$ is denoted by
${V^{\star}_{{\delta}}({{\widehat{{\Omega}}}_{\delta}})}$ with
$(\cdot,\cdot)_{{\widehat{{\Omega}}}_{\delta}}$ being the duality pairing
which, for functions vanishing on ${{\Omega}^{D}_{{\mathcal{I}}_{\delta}}}$,
is also used to denote the equivalent
$L^{2}({\Omega}\cup{{\Omega}^{N}_{{\mathcal{I}}_{\delta}}})$ inner product.

For convenience, we also define the energy seminorm

$|v|_{{V_{c,{\delta}}({{\widehat{{\Omega}}}_{\delta}})}}=\biggl{(}\frac{1}{2}\int_{{\widetilde{{\Omega}}_{\delta}}}(v({\bm{y}})-v({\bm{x}}))^{2}{\gamma_{\delta}}{({\bm{x}},{\bm{y}})}{d{\bm{y}}d{\bm{x}}}\biggr{)}^{1/2},$
(53)

which can be a full norm on
${V_{c,{\delta}}({{\widehat{{\Omega}}}_{\delta}})}$ as shown later.

######  Remark 3.2.

A natural question is to ask what assumptions on
${{\Omega}^{D}_{{\mathcal{I}}_{\delta}}}$ and
${{\Omega}^{N}_{{\mathcal{I}}_{\delta}}}$ and ${g_{D}^{\delta}}$ and
${g_{N}^{\delta}}$ are needed to recover the appropriate local limit. For
example, under what assumptions on ${{\Omega}^{D}_{{\mathcal{I}}_{\delta}}}$
and ${{\Omega}^{N}_{{\mathcal{I}}_{\delta}}}$, one can have bounded extension
of functions ${V_{c,{\delta}}({{\widehat{{\Omega}}}_{\delta}})}$ with respect
to the energy, in the same form but over some larger domain, independent of
$\delta$ for $\delta$ small. Sometimes, it is possible that the above
extension cannot be made: for example, if
${{\Omega}^{D}_{{\mathcal{I}}_{\delta}}}$ is an inner layer surrounded by
${{\Omega}^{N}_{{\mathcal{I}}_{\delta}}}$. For an attempt to put assumptions
on ${{\Omega}^{D}_{{\mathcal{I}}_{\delta}}}$,
${{\Omega}^{N}_{{\mathcal{I}}_{\delta}}}$, ${g_{D}^{\delta}}$ and
${g_{N}^{\delta}}$, we refer to the discussion in Section 4.2.

#### Compactness and equivalent norms

The work of Bourgain, Brezis and Mironecu [9] characterizes the limiting space
of the energy space ${V_{{\delta}}({{\widehat{{\Omega}}}_{\delta}})}$ as
$\delta\to 0$ when the kernel ${\gamma_{\delta}}$ satisfies (18) and (20).
Compactness is gained through taking the limit $\delta\to 0$. The original
result in [9] requires additionally that ${\gamma}(t)t^{2}$ is a nonincreasing
function for $t\in(0,1)$. It was found later that the assumption could be
replaced with ${\gamma}(t)$ being nonincreasing instead, see e.g., discussions
in [54, 73] and [34]. [61] showed a different proof of the Bourgain-Brezis-
Mironescu compactness result which removed the nonincreasing assumption on
${\gamma}$ for $d\geq 2$. We quote the compactness result below.

######  Lemma 1.

Assume that ${\delta}_{n}\to 0$ and the kernels
$\\{\gamma_{\delta_{n}}\\}_{n\in\mathbb{N}_{+}}$ satisfy (18) and (20). Let
${\Omega}_{1}\subset{{\mathbb{R}}^{d}}$ be a bounded Lipschitz domain. If
$\\{v_{n}\\}\subset L^{2}({\Omega}_{1})$ is a bounded sequence such that

$\int_{{\Omega}_{1}}\int_{{\Omega}_{1}}\gamma_{\delta_{n}}({\bm{x}},{\bm{y}})(v_{n}({\bm{y}})-v_{n}({\bm{x}}))^{2}{d{\bm{x}}}{d{\bm{y}}}\leq
C_{0},$

then $\\{v_{n}\\}$ is relatively compact in $L^{2}({\Omega}_{1})$ and any
limit function $v$ belongs to $H^{1}({\Omega}_{1})$ with

$\|v\|_{H^{1}({\Omega}_{1})}\leq C_{0}.$

######  Remark 3.3.

It is possible to have more general compactness results for a sequence of
kernels $\\{\gamma_{\delta_{n}}\\}_{n\in\mathbb{N}_{+}}$ converging to a limit
$\gamma_{\delta}$, as $\delta_{n}\to\delta$, under suitable conditions on
these kernels and the corresponding energy spaces
${V_{c,{\delta}_{n}}({{\widehat{{\Omega}}}_{{\delta}_{n}}})}$ and
${V_{c,{\delta}}({{\widehat{{\Omega}}}_{\delta}})}$. We refer to [73, 34].

######  Remark 3.4.

Note that by classical results of fractional Sobolev spaces, for a kernel
$\gamma_{\delta}$ that satisfies (22), we have the pre-compactness of
${V_{c,{\delta}}({{\widehat{{\Omega}}}_{\delta}})}$ in
$L^{2}({\widehat{{\Omega}}})$ for $\delta>0$.

Using the compactness result, we can prove a general Poincaré inequality [30,
52, 28, 51] for kernels satisfying either (18)-(20) or (22) as $\delta\to 0$
and $\delta\to\infty$ respectively.

######  Lemma 2.

There exists a constant $C>0$, independent of ${\delta}$, such that

$\|v\|_{L^{2}({{\widehat{{\Omega}}}_{\delta}})}\leq
C|v|_{{V_{c,{\delta}}({{\widehat{{\Omega}}}_{\delta}})}},\quad\forall\;v\in{V_{c,{\delta}}({{\widehat{{\Omega}}}_{\delta}})}.$
(54)

Lemma 2 unifies the classical version of the Poincaré inequality for the usual
energy function spaces associated with the local Laplacian, the fractional and
the nonlocal versions.

######  Remark 3.5.

We see that $|\cdot|_{{V_{c,{\delta}}({{\widehat{{\Omega}}}_{\delta}})}}$ is a
norm on ${V_{c,{\delta}}({{\widehat{{\Omega}}}_{\delta}})}$ equivalent to the
norm $\|\cdot\|_{{V_{c,{\delta}}({{\widehat{{\Omega}}}_{\delta}})}}$ uniformly
in ${\delta}$ for kernels specified in (20) and (22) as $\delta\to 0$ and
$\delta\to\infty$ respectively. Correspondingly, we may view
${V_{c,{\delta}}({{\widehat{{\Omega}}}_{\delta}})}$ as a Hilbert space
equipped with the inner product

$\displaystyle(u,v)_{{V_{c,{\delta}}({{\widehat{{\Omega}}}_{\delta}})}}=\frac{1}{2}\int_{{\widetilde{{\Omega}}_{\delta}}}(v({\bm{y}})-v({\bm{x}}))(u({\bm{y}})-u({\bm{x}})){\gamma_{\delta}}{({\bm{x}},{\bm{y}})}{d{\bm{y}}d{\bm{x}}}$
(55)

for all $u,v\in{V_{c,{\delta}}({{\widehat{{\Omega}}}_{\delta}})}$.

For more detailed discussions of the conditions on the kernels and rigorous
proofs of the properties on these spaces, we refer to [54, 55, 28].

For any subdomain
${\Omega}^{\prime}\subset{{\Omega}_{{\mathcal{I}}_{\delta}}}$ with a nonzero
volume (i.e., $d$ dimensional measure), we define the space
${V_{t}({\Omega}^{\prime})}$,

${V_{t}({\Omega}^{\prime})}=\\{v=w|_{{\Omega}^{\prime}}\;\text{for
some}\;w\in{V_{{\delta}}({{\widehat{{\Omega}}}_{\delta}})}\\},$ (56)

which involves restrictions to the domain ${\Omega}^{\prime}$ having nonzero
volume in ${{\mathbb{R}}^{d}}$. A norm on ${V_{t}({\Omega}^{\prime})}$ can be
defined by

$\displaystyle\|v\|_{{V_{t}({\Omega}^{\prime})}}$
$\displaystyle=\inf\\{\|w\|_{{V_{{\delta}}({{\widehat{{\Omega}}}_{\delta}})}}\;|\;w\in{V_{{\delta}}({{\widehat{{\Omega}}}_{\delta}})},w|_{{\Omega}^{\prime}}=v.\\}$
(57)

######  Remark 3.6.

One can view ${V_{t}({\Omega}^{\prime})}$, induced by
${V_{{\delta}}({{\widehat{{\Omega}}}_{\delta}})}$, as a nonlocal analogue of
the trace space $H^{1/2}(S)$ induced by $H^{1}({\Omega})$ for
$S\subset\partial{\Omega}$. Some rigorous characterizations of the trace
spaces for special classes of kernels can be found in [38].

We are particularly interested in the spaces
${V_{t}({{\Omega}^{D}_{{\mathcal{I}}_{\delta}}})}$ and
${V_{t}({{\Omega}^{N}_{{\mathcal{I}}_{\delta}}})}$ and their dual spaces
${V_{t}^{\star}({{\Omega}^{D}_{{\mathcal{I}}_{\delta}}})}$ and
${V_{t}^{\star}({{\Omega}^{N}_{{\mathcal{I}}_{\delta}}})}$ corresponding to
cases ${\Omega}^{\prime}={{\Omega}^{D}_{{\mathcal{I}}_{\delta}}}$ and
${\Omega}^{\prime}={{\Omega}^{N}_{{\mathcal{I}}_{\delta}}}$ respectively. We
use
${{\langle{{g_{N}^{\delta}}},{v}\rangle}_{{\Omega}^{N}_{{\mathcal{I}}_{\delta}}}}$
to denote the inner product of ${g_{N}^{\delta}}$ and $v$ in
$L^{2}({{\Omega}^{N}_{{\mathcal{I}}_{\delta}}})$.

###  3.2 Weak formulations and well-posedness

For illustration, we consider the case $F({\bm{x}},u)=-q(u)+f({\bm{x}})$ where
$q(u)=Q^{\prime}(u)$ with $Q=Q(u)=\alpha|u|^{m}/m$ with some
$m\in(2,\min\\{4,2d/(d-2)\\})$ and $\alpha\geq 0$. We note that the problem
reduces to a linear one if $\alpha=0$. For any $u\in L^{m}({\Omega})$, we
define a linear functional

${\mathcal{Q}}(u,v)=\int_{\Omega}q(u({\bm{x}}))v({\bm{x}})d{\bm{x}},\quad\forall
v\in L^{m}({\Omega}).$

Following [28], we consider a weak formulation of problem (12) with
${{\mathcal{B}}_{\delta}}u={g_{\delta}}$. Let ${g_{D}^{\delta}}$ and
${g_{N}^{\delta}}$ denote the restriction of ${g_{\delta}}$ on
${{\Omega}^{D}_{{\mathcal{I}}_{\delta}}}$ and
${{\Omega}^{N}_{{\mathcal{I}}_{\delta}}}$ respectively for domains that have
nonzero volume ($d$ dimensional measure). We define

${\widehat{f}}({\bm{x}})=\begin{cases}f({\bm{x}}),&\quad{\bm{x}}\in{\Omega},\\\
0,&\quad{\bm{x}}\in{{\Omega}^{D}_{{\mathcal{I}}_{\delta}}},\\\
{g_{N}^{\delta}}({\bm{x}}),&\quad{\bm{x}}\in{{\Omega}^{N}_{{\mathcal{I}}_{\delta}}}.\end{cases}$
(58)

We assume that
${\widehat{f}}\in{V^{\star}_{{\delta}}({{\widehat{{\Omega}}}_{\delta}})}$,
which also implies
${g_{N}^{\delta}}\in{V_{t}^{\star}({{\Omega}^{N}_{{\mathcal{I}}_{\delta}}})}$.
Moreover, we assume that
${g_{D}^{\delta}}\in{V_{t}({{\Omega}^{D}_{{\mathcal{I}}_{\delta}}})}$. By the
definition of ${V_{t}({{\Omega}^{D}_{{\mathcal{I}}_{\delta}}})}$, we can
define an extension of ${g_{D}^{\delta}}$ on
${{\widehat{{\Omega}}}_{\delta}}$, denoted by $w_{D}^{\delta}$, such that

$\displaystyle
w_{D}^{\delta}\in{V_{{\delta}}({{\widehat{{\Omega}}}_{\delta}})},\quad
w_{D}^{\delta}\mid_{{{\Omega}^{D}_{{\mathcal{I}}_{\delta}}}}={g_{D}^{\delta}},\quad\|w_{D}^{\delta}\|_{{V_{{\delta}}({{\widehat{{\Omega}}}_{\delta}})}}\leq
C\|{g_{D}^{\delta}}\|_{{V_{t}({{\Omega}^{D}_{{\mathcal{I}}_{\delta}}})}},$
(59)

for some constant $C>0$, independent of $\delta$. Without loss of generality,
we can set $C=2$. To deal with the special nonlinear term, we make an
additional assumption that $w_{D}^{\delta}$ also satisfies

$\displaystyle\alpha w_{D}^{\delta}\in\alpha
L^{m}({\Omega}),\quad\quad{\mathcal{Q}}(w_{D}^{\delta},w_{D}^{\delta})\leq\tilde{q}(\|{g_{D}^{\delta}}\|_{{V_{t}({{\Omega}^{D}_{{\mathcal{I}}_{\delta}}})}}),$
(60)

for a given smooth function $\tilde{q}=\tilde{q}(r)$. Here we use the
abbreviation $\alpha L^{m}({\Omega})$ as the space of all functions $v$ such
that $\alpha v\in L^{m}({\Omega})$, i.e., $\alpha L^{m}({\Omega})$ can include
all functions if $\alpha=0$ but $\alpha L^{m}({\Omega})=L^{m}({\Omega})$ if
$\alpha\neq 0$.

The weak formulation can then be stated as follows: given
${g_{D}^{\delta}}\in{V_{t}({{\Omega}^{D}_{{\mathcal{I}}_{\delta}}})}$,
${\widehat{f}}\in{V^{\star}_{{\delta}}({{\widehat{{\Omega}}}_{\delta}})}$, and
$w_{D}^{\delta}$ satisfying (59) and (60), find $u$ such that
$u-w_{D}^{\delta}\in{V_{c,{\delta}}({{\widehat{{\Omega}}}_{\delta}})}\cap\alpha
L^{m}({\Omega})$ and

${{{\mathcal{A}}_{\delta}}}(u,v)+{{\mathcal{Q}}(u,v)}={({\widehat{f}},v)_{{\widehat{{\Omega}}}_{\delta}}},\quad\forall
v\in{V_{c,{\delta}}({{\widehat{{\Omega}}}_{\delta}})}{,}$ (61)

where the bilinear form ${{{\mathcal{A}}_{\delta}}}(\cdot,\cdot)$ is given by

$\displaystyle{{{\mathcal{A}}_{\delta}}}(u,v)=(u,v)_{{V_{c,{\delta}}({{\widehat{{\Omega}}}_{\delta}})}},\quad\forall\>u,v\in{V_{{\delta}}({{\widehat{{\Omega}}}_{\delta}})},$
(62)

and the linear functional

${({\widehat{f}},v)_{{\widehat{{\Omega}}}_{\delta}}}=\int_{{\Omega}}f({\bm{x}})v({\bm{x}}){d{\bm{x}}}+\int_{{{\Omega}^{N}_{{\mathcal{I}}_{\delta}}}}{g_{N}^{\delta}}({\bm{x}})v({\bm{x}}){d{\bm{x}}}\,.$
(63)

Note that if
${{\Omega}^{D}_{{\mathcal{I}}_{\delta}}}={{\Omega}_{{\mathcal{I}}_{\delta}}}$,
then for any $v\in{V_{c,{\delta}}({{\widehat{{\Omega}}}_{\delta}})}$, we have

$\displaystyle{{{\mathcal{A}}_{\delta}}}(u,v)=(u,v)_{{V_{c,{\delta}}({{\widehat{{\Omega}}}_{\delta}})}}$
$\displaystyle=\frac{1}{2}\int_{{\widetilde{{\Omega}}_{\delta}}}(v({\bm{y}})-v({\bm{x}}))(u({\bm{y}})-u({\bm{x}})){\gamma_{\delta}}{({\bm{x}},{\bm{y}})}{d{\bm{y}}d{\bm{x}}}$
(64)
$\displaystyle=\frac{1}{2}\int_{{{\widehat{{\Omega}}}_{\delta}}}\int_{{{\widehat{{\Omega}}}_{\delta}}}(v({\bm{y}})-v({\bm{x}}))(u({\bm{y}})-u({\bm{x}})){\gamma_{\delta}}{({\bm{x}},{\bm{y}})}{d{\bm{y}}d{\bm{x}}}$

The weak formulation is equivalent to a minimization problem: Given
${g_{D}^{\delta}}\in{V_{t}({{\Omega}^{D}_{{\mathcal{I}}_{\delta}}})}$,
${\widehat{f}}\in{V^{\star}_{{\delta}}({{\widehat{{\Omega}}}_{\delta}})}$, and
$w_{D}^{\delta}$ satisfying (59) and (60), find $u$ that minimizes

$\min_{v\in
w_{D}^{\delta}+{V_{c,{\delta}}({{\widehat{{\Omega}}}_{\delta}})}\cap\alpha
L^{m}({\Omega})}E(v):={\frac{1}{2}{{\mathcal{A}}_{\delta}}}(v,v)+\frac{1}{m}{\mathcal{Q}}(v,v)-{({\widehat{f}},v)_{{\widehat{{\Omega}}}_{\delta}}},$
(65)

######  Theorem 3.

The problem (61) has a unique solution $u\in
w_{D}^{\delta}+{V_{c,{\delta}}({{\widehat{{\Omega}}}_{\delta}})}\cap\alpha
L^{m}({\Omega})$, for any given data ${\widehat{f}}$, ${g_{D}^{\delta}}$ and
$w_{D}^{\delta}$ satisfying the conditions specified earlier. Moreover, there
is a constant $C>0$, independent of ${\delta}$, such that the following a
priori estimates for the solution $u$ are satisfied:

$\|u\|_{V_{{\delta}}({{\widehat{{\Omega}}}_{\delta}})}^{2}+{\mathcal{Q}}(u,u)\leq
C[\|{\widehat{f}}\|_{V^{\star}_{{\delta}}({{\widehat{{\Omega}}}_{\delta}})}^{2}+\|{g_{D}^{\delta}}\|_{V_{t}({{\Omega}^{D}_{{\mathcal{I}}_{\delta}}})}^{2}+\tilde{q}(\|{g_{D}^{\delta}}\|_{V_{t}({{\Omega}^{D}_{{\mathcal{I}}_{\delta}}})})].$
(66)

###### Proof.

Using the Poincaré inequality in Lemma 2, we see that the functional is
coercive and continuous in the set
$w_{D}^{\delta}+{V_{c,{\delta}}({{\widehat{{\Omega}}}_{\delta}})}\cap\alpha
L^{m}({\Omega})$ with respect to the norm
$\|\cdot\|_{L^{2}_{0}({\Omega})}+\alpha\|\cdot\|_{L^{m}({\Omega})}$. Thus,
there exists a minimizing sequence which remains uniformly bounded in
$w_{D}^{\delta}+{V_{c,{\delta}}({{\widehat{{\Omega}}}_{\delta}})}\cap\alpha
L^{m}({\Omega})$. One can take a uniformly bounded and weakly convergent
sequence $v_{n}=w_{n}+w_{D}^{\delta}$,
$w_{n}\in{V_{c,{\delta}}({{\widehat{{\Omega}}}_{\delta}})}\cap\alpha
L^{m}({\Omega})$ such that

$\lim_{n\to\infty}E(v_{n})=\inf_{v\in
w_{D}^{\delta}+{V_{c,{\delta}}({{\widehat{{\Omega}}}_{\delta}})}\cap\alpha
L^{m}({\Omega})}E(v)>-\infty.$

Denote $u$ the limit of $v_{n}$, then $u\in
w_{D}^{\delta}+{V_{c,{\delta}}({{\widehat{{\Omega}}}_{\delta}})}\cap\alpha
L^{m}({\Omega})$, and we may use the weak lower semicontinuity of the
functional $E=E(v)$ to get

$E(u)\leq\underline{\lim}_{n\to\infty}E(v_{n})=\inf_{v\in
w_{D}^{\delta}+{V_{c,{\delta}}({{\widehat{{\Omega}}}_{\delta}})}\cap\alpha
L^{m}({\Omega})}E(v),$

which gives $u$ as the minimizer of $E(v)$ in
$w_{D}^{\delta}+{V_{c,{\delta}}({{\widehat{{\Omega}}}_{\delta}})}\cap\alpha
L^{m}({\Omega})$. One can easily see that the weak formulation is simply the
Euler-Lagrange equation. We thus have the existence of the solution. Since
$E(v)$ is strictly convex, we also get the uniqueness of the solution. The a
priori estimates can be obtained from the weak formulation. Indeed,

${{{\mathcal{A}}_{\delta}}}(u,u-w_{D}^{\delta})+{{\mathcal{Q}}(u,u-w_{D}^{\delta})}=(\widehat{f},u-w_{D}^{\delta})_{{{\widehat{{\Omega}}}_{\delta}}}.$

Then Young’s inequality implies

$\displaystyle\frac{1}{\alpha}Q(u,u-w_{D}^{\delta})$
$\displaystyle=\int_{{\Omega}}u^{m}({\bm{x}})d{\bm{x}}-\int_{{\Omega}}u^{m-1}({\bm{x}})w_{D}^{\delta}({\bm{x}})d{\bm{x}}$
$\displaystyle\geq\int_{{\Omega}}u^{m}({\bm{x}})d{\bm{x}}-\int_{{\Omega}}\frac{m-1}{m}u^{m}({\bm{x}})+\frac{1}{m}(w_{D}^{\delta})^{m}({\bm{x}})d{\bm{x}}$
$\displaystyle=\frac{1}{m}\int_{{\Omega}}u^{m}({\bm{x}})d{\bm{x}}-\frac{1}{m}\int_{{\Omega}}(w_{D}^{\delta})^{m}({\bm{x}})d{\bm{x}}.$

Note that for the linear case with $\alpha=0$, we can also use the Lax–Milgram
theorem to show the well-posedness of the problem (61). ∎

######  Remark 3.7 (a pure nonlocal Neumann problem).

For the case
${{\Omega}^{N}_{{\mathcal{I}}_{\delta}}}={{\Omega}_{{\mathcal{I}}_{\delta}}}$
and $\alpha=0$, that is, a pure linear Neumann problem, due to the
compatibility condition imposed on
${V_{c,{\delta}}({{\widehat{{\Omega}}}_{\delta}})}$, we can recover the
original strong form (12) from the linear functional
${({\widehat{f}},v)_{\widehat{{\Omega}}}}$ with the following compatibility
condition:

$\displaystyle\int_{{{\widehat{{\Omega}}}_{\delta}}}{\widehat{f}}({\bm{x}}){d{\bm{x}}}=\int_{{\Omega}}f({\bm{x}}){d{\bm{x}}}+\int_{{{\Omega}_{{\mathcal{I}}_{\delta}}}}{g_{N}^{\delta}}({\bm{x}}){d{\bm{x}}}=0.$
(67)

Meanwhile, for the nonlinear problem (27), the condition is equivalently (28).
Similarly, a compatibility condition is also required for the “regional”
nonlocal diffusion problem given in (31) or the equivalent form in (33):

$\displaystyle\int_{{\Omega}}F({\bm{x}},u({\bm{x}}))d{\bm{x}}=\int_{{\Omega}_{1}}F({\bm{x}},u({\bm{x}}))d{\bm{x}}+\int_{{\Omega}_{2}}G({\bm{x}},u({\bm{x}}))d{\bm{x}}=0$
(68)

where ${\Omega}_{1}$ and ${\Omega}_{2}$ represent a decomposition of
${\Omega}$ as in (33). The equation (68) follows easily from the symmetry
assumption on the nonlocal interaction kernel.

######  Remark 3.8.

From the discussion here, we see that in the weak formulation, different parts
of the data ${\widehat{f}}$ play different roles, in particular when
considering the limiting regimes of $\delta\to 0$ or $\delta\to\infty$. That
is, $f$ symbolically corresponds to the right-hand side of the equation while
${g_{N}^{\delta}}$ corresponds to the Neumann data.

######  Remark 3.9.

We note that other ways of defining a nonlocal Neumann problem can also be
used. For additional discussions about nonlocal Neumann problems, see e.g.
[17, 4, 28, 54, 55, 63, 49, 69, 23, 24]. In addition, we may also consider
problems associated with the regional nonlocal and fractional Laplacian
defined by (32) and(29) on the domain ${\Omega}$. In particular, the bilinear
form can be modified as

$\displaystyle{\mathcal{A}}_{\delta,{\Omega}}(u,v)=(u,v)_{{V_{c,{\delta}}({\Omega})}}=\frac{1}{2}\int_{{\Omega}}\int_{{\Omega}}(v({\bm{y}})-v({\bm{x}}))(u({\bm{y}})-u({\bm{x}})){\gamma_{\delta}}{({\bm{x}},{\bm{y}})}{d{\bm{y}}d{\bm{x}}}$
(69)

for all $u,v\in{V_{c,{\delta}}({\Omega})}$.

######  Remark 3.10.

Recall the Robin type constraint specified by (24) with (25), we can get
similar well-posedness in such a case by modifying the bilinear form to

$\displaystyle{{{\mathcal{A}}_{\delta}}}(u,v)=(u,v)_{{V_{c,{\delta}}({{\widehat{{\Omega}}}_{\delta}})}}+({{\mathcal{D}}_{\delta}}u,v)_{{{\Omega}_{{\mathcal{I}}_{\delta}}}},\quad\forall\>u,v\in{V_{{\delta}}({{\widehat{{\Omega}}}_{\delta}})},$
(70)

For given ${\delta}>0$, if
$({{\mathcal{D}}_{\delta}}u,u)_{{{\Omega}_{{\mathcal{I}}_{\delta}}}}=0$ for
$u\in{V_{{\delta}}({{\widehat{{\Omega}}}_{\delta}})}$ holds only if
$u\mid_{{\Omega}_{{\mathcal{I}}_{\delta}}}=0$, then the coercivity of the
modified bilinear form can also be derived (e.g., by following similar
discussions on nonlocal Poincaré inequality given in [54]).

To conclude this section, we note that one can establish the maximum principle
for linear nonlocal diffusion problems as for the local case; see discussions
in [41, 33].

##  4 Special examples and limiting cases

In this section, we consider some special cases, including nonlocal models
with an integrable kernel that are of particular interest in applications like
peridynamics. Then, we consider the local ($\delta\to 0$) and fractional
($\delta\to\infty$) limits of nonlocal diffusion models parameterized by
$\delta>0$. This again is to demonstrate the bridging role of the nonlocal
models with a finite $\delta>0$, i.e., as illustrated in Figure. 4.1, which
generalizes some similar pictures in [75] for homogeneous Dirichlet problems.

${\Omega}$${{\Omega}_{{\mathcal{I}}_{0}}}={\partial{\Omega}}$$-\mathcal{L}_{0}u_{0}$=$\,{f}$Local${{\mathcal{B}}_{0}}{u}_{0}$=$\,{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}g_{0}}$$0\leftarrow\delta$(20)${\Omega}$${{\Omega}_{\mathcal{I}_{\delta}}}$$-{{\mathcal{L}}_{\delta}}{u}_{\delta}$=$\,{f}$Nonlocal${{\mathcal{B}}_{\delta}}{u}_{\delta}$
=$\,{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}g_{\delta}}$$\delta\rightarrow\infty$(22)${\Omega}$${{\Omega}_{{\mathcal{I}}_{\infty}}}={{\Omega}_{c}}$${{\mathcal{B}}_{\infty}^{s}}{u}_{\infty}$=$\,{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}g_{\infty}}$$-$${{\mathcal{L}}_{\infty}^{s}}{u}_{\infty}$=$\,{f}$
Fractional
Fig. 4.1:  Limits of nonlocal models with a finite interaction radius $\delta$
as $\delta\to 0$ (local limit) with a kernel given by (20) and as
$\delta\to\infty$ (fractional limit) with a kernel given by (22).

###  4.1 Nonlocal Laplacian with integrable kernels

We consider the case with a kernel satisfying

$\int_{{{\mathbb{R}}^{d}}}{\gamma_{\delta}}{({\bm{x}},{\bm{y}})}d{\bm{y}}={{\delta}^{d}}C_{d}({\delta})\int_{{B_{1}({\bf
0})}}\gamma(|{\bm{z}}|)\,d{\bm{z}}<\infty.$ (71)

In this case, for any given $\delta>0$, ${{\mathcal{L}}_{\delta}}$ and
${{\mathcal{N}}_{\delta}}$ become bounded operators on the $L^{2}$ spaces for
functions defined on respective domains. One can see that the functions spaces
are essentially the $L^{2}$ spaces on the corresponding domains [53]. Then for
the problem,

$\begin{cases}-{{\mathcal{L}}_{\delta}}u=f({\bm{x}}),&\quad\forall\,{\bm{x}}\in{\Omega},\\\
u={g_{D}^{\delta}}({\bm{x}}),&\quad\forall\,{\bm{x}}\in{{\Omega}^{D}_{{\mathcal{I}}_{\delta}}},\\\
{{\mathcal{N}}_{\delta}}u={g_{N}^{\delta}}({\bm{x}}),&\quad\forall\,{\bm{x}}\in{{\Omega}^{N}_{{\mathcal{I}}_{\delta}}},\end{cases}$
(72)

the basic existence of solution in $L^{2}({{\widehat{{\Omega}}}_{\delta}})$
can be obtained, similar to earlier discussion, for $f\in L^{2}({\Omega})$,
${g_{D}^{\delta}}\in L^{2}({{\Omega}^{D}_{{\mathcal{I}}_{\delta}}})$ and
${g_{N}^{\delta}}\in L^{2}({{\Omega}^{N}_{{\mathcal{I}}_{\delta}}})$.

We present this special case to highlight the fact that the solutions to the
nonlocal diffusion problem may not share the smoothing properties enjoyed by
the solutions to typical elliptic PDEs; that is, the solutions may not have a
higher order of differentiability than the data.

###  4.2 Local limit

We present some results concerning the local limits. We begin with a formal
discussion. For example, equation (61) is a nonlocal analog of the weak
formulation corresponding to (3) with ${{\mathcal{B}}_{0}}u=u$ on
${\partial{\Omega}^{D}}$ and ${{\mathcal{B}}_{0}}u=\frac{\partial
u}{\partial\mathbf{n}}$ on ${\partial{\Omega}^{N}}$ with the weak form given
by

$\int_{\Omega}\nabla u{\,\cdot\,}\nabla
v{d{\bm{x}}}{+{\mathcal{Q}}(u,v)}=\int_{\Omega}fv{d{\bm{x}}}+\int_{{\partial{\Omega}^{N}}}g^{0}_{N}v{d{\bm{x}}}.$
(73)

For a pure linear Neumann problem with
${\partial{\Omega}^{N}}={\partial{\Omega}}$, i.e., $\alpha=0$ in (73) the
functions $f$ and $g_{N}$ satisfy the conventional compatibility condition

${\int_{\Omega}f{d{\bm{x}}}+\int_{\partial{\Omega}}g^{0}_{N}{d{\bm{x}}}=0.}$

Next, we present some rigorous results concerning the limit of $\delta\to 0$
of (12) for kernels satisfying (18) and (20). To properly discuss the local
limit, we need to make several assumptions on the boundary domains and
boundary data.

######  Assumption 4.1.

We first make the following assumptions on the domains.

  * $\bullet$

The domains ${{\Omega}^{D}_{{\mathcal{I}}_{\delta}}}$ and
${{\Omega}^{N}_{{\mathcal{I}}_{\delta}}}$ are monotone decreasing sets as
${\delta}\to 0$ and
$\overline{{{\Omega}^{D}_{{\mathcal{I}}_{\delta}}}}\supset{\partial{\Omega}^{D}}$
and
$\overline{{{\Omega}^{N}_{{\mathcal{I}}_{\delta}}}}\supset{\partial{\Omega}^{N}}$.

  * $\bullet$

Moreover, for any ${\delta}_{1}>{\delta}_{2}>0$, we assume that
$({\Omega}^{D}_{\mathcal{I}_{{\delta}_{1}}}\backslash{\Omega}^{D}_{\mathcal{I}_{{\delta}_{2}}})\cap{\Omega}_{\mathcal{I}_{{\delta}_{2}}}=\emptyset$.

Next, we present the assumptions on the data.

  * $\bullet$

For a constant ${\delta}_{0}>0$, we assume that for all
${\delta}<{\delta}_{0}$, the function $w_{D}^{\delta}$ in (59) and(60) is
extended to ${\widehat{{\Omega}}}_{{\delta}_{0}}$ with the extension bounded
in $V_{c,{\delta}_{0}}({\widehat{{\Omega}}}_{{\delta}_{0}})\cap\alpha
L^{m}({\Omega})$ and $w_{D}^{\delta}$ converges in
$L^{2}({\widehat{{\Omega}}}_{{\delta}_{0}})$ to $w_{D}^{0}\in
H^{1}({\widehat{{\Omega}}}_{{\delta}_{0}})$ as ${\delta}\to 0$, where
$w_{D}^{0}|_{{\partial{\Omega}^{D}}}=g_{D}^{0}$.

  * $\bullet$

We also assume that the Neumann data ${g_{N}^{\delta}}$, as a function defined
on ${{\widehat{{\Omega}}}_{\delta}}$, is uniformly bounded in the dual space
of the energy space. Moreover,

$\displaystyle\lim_{{\delta}\to
0}\int_{{{\Omega}^{N}_{{\mathcal{I}}_{\delta}}}}{g_{N}^{\delta}}({\bm{x}})v({\bm{x}}){d{\bm{x}}}=\int_{{\partial{\Omega}^{N}}}g_{N}^{0}({\bm{x}})v({\bm{x}}){d{\bm{x}}}\,,\quad\forall
v\in C^{\infty}({{\mathbb{R}}^{d}}).$ (74)

######  Remark 4.2.

Let us note that under the assumption on $m$ and the Sobolev imbedding
theorem, we have $\alpha w_{D}^{0}\in\alpha L^{m}({\Omega})$.

######  Remark 4.3.

Notice the special geometric assumptions on the domains
${{\Omega}^{D}_{{\mathcal{I}}_{\delta}}}$ and
${{\Omega}^{N}_{{\mathcal{I}}_{\delta}}}$ in the above discussions. In
particular, we require
$\\{\overline{{{\Omega}^{D}_{{\mathcal{I}}_{\delta}}}}\\}_{{\delta}>0}$ and
$\\{\overline{{{\Omega}^{N}_{{\mathcal{I}}_{\delta}}}}\\}_{{\delta}>0}$ to be
sequences of nested sets that decrease monotonically to
${\partial{\Omega}^{D}}$ and ${\partial{\Omega}^{N}}$, respectively, as
$\delta\to 0$. These assumptions rule out the more complicated situations
where ${{\Omega}^{D}_{{\mathcal{I}}_{\delta}}}$ or
${{\Omega}^{N}_{{\mathcal{I}}_{\delta}}}$ may not be attached to the boundary
$\partial{\Omega}$. Some numerical experiments are provided in cases these
geometric assumptions are violated.

With these assumptions, we present the convergence theorem for $\delta\to 0$.

######  Theorem 4.4.

Let $u_{\delta}$ and $u_{0}$ be the weak solutions to the nonlocal and local
boundary value problems, respectively. Assume that
$\|{\widehat{f}}\|_{V^{\star}_{{\delta}}({{\widehat{{\Omega}}}_{\delta}})}$
and $\|{g_{D}^{\delta}}\|_{V_{t}({{\Omega}^{D}_{{\mathcal{I}}_{\delta}}})}$
are bounded uniformly in ${\delta}$, then we have

$\|u_{\delta}-u_{0}\|_{L^{2}({\Omega})}\to 0\,,$

as ${\delta}\to 0$.

###### Proof.

From the well-posedness theorem, it is easy to see that
$\|u_{\delta}\|_{{V_{{\delta}}({{\widehat{{\Omega}}}_{\delta}})}}$,
$\|u_{\delta}-w_{D}^{\delta}\|_{{V_{{\delta}}({{\widehat{{\Omega}}}_{\delta}})}}$
and ${\mathcal{Q}}(u_{\delta},u_{\delta})$ are uniformly bounded and
$(u_{\delta}-w_{D}^{\delta})|_{{{\Omega}^{D}_{{\mathcal{I}}_{\delta}}}}=0$.
Take ${\delta}_{0}>0$ and let
${\Omega}_{1}:=\overline{{\Omega}\cup{\Omega}^{D}_{\mathcal{I}_{{\delta}_{0}}}}$.
By the assumptions on the boundary sets, for any ${\delta}<{\delta}_{0}$, we
can always do a zero extension of the function $u_{\delta}-w_{D}^{\delta}$
onto ${\Omega}_{1}$, and

$\begin{split}&\int_{{\Omega}_{1}}\int_{{\Omega}_{1}}{\gamma_{\delta}}({\bm{x}},{\bm{y}})\left((u_{\delta}-w_{D}^{\delta})({\bm{y}})-(u_{\delta}-w_{D}^{\delta})({\bm{x}})\right)^{2}{d{\bm{y}}}{d{\bm{x}}}\\\
\leq&\int_{({\Omega}\cup{{\Omega}^{D}_{{\mathcal{I}}_{\delta}}})^{2}\backslash({{\Omega}^{D}_{{\mathcal{I}}_{\delta}}})^{2}}{\gamma_{\delta}}({\bm{x}},{\bm{y}})\left((u_{\delta}-w_{D}^{\delta})({\bm{y}})-(u_{\delta}-w_{D}^{\delta})({\bm{x}})\right)^{2}{d{\bm{y}}}{d{\bm{x}}}\\\
\end{split}$

We can then use Lemma 1 to conclude that any $L^{2}({\Omega}_{1})$-limit (up
to a subsequence) of $u_{\delta}-w_{D}^{\delta}$ as ${\delta}\to 0$ belongs to
$H^{1}({\Omega}_{1})$. Let $v^{\ast}$ denote the limit function and
$u^{\ast}:=v^{\ast}+w_{D}^{0}$. By the assumption on the Dirichlet data,
$u^{\ast}$ is the $L^{2}({\Omega}_{1})$-limit of $u_{\delta}$. We want to show
$u^{\ast}=u_{0}$. First, since
$(u_{\delta}-w_{D}^{\delta})|_{{\Omega}^{D}_{\mathcal{I}_{{\delta}_{0}}}}=0$
for all ${\delta}<{\delta}_{0}$, we then have
$v^{\ast}|_{{\Omega}^{D}_{\mathcal{I}_{{\delta}_{0}}}}=0$. This implies that
$u^{\ast}\in H^{1}({\Omega})$ and
$u^{\ast}|_{{\partial{\Omega}^{D}}}=w_{D}^{0}|_{{\partial{\Omega}^{D}}}=g_{D}^{0}$.
Now we only need to show that $u^{\ast}$ satisfies the weak form (73) for any
$v\in C^{\infty}(\overline{{\Omega}})$ with $v|_{{\partial{\Omega}^{D}}}=0$.
Assume that the extension of $v$ is made such that $v\in
C^{\infty}(\overline{{\Omega}\cup{\Omega}_{\mathcal{I}_{{\delta}_{0}}}})$ and
$v|_{{\Omega}^{D}_{\mathcal{I}_{{\delta}_{0}}}}=0$. Since $u_{\delta}$ is the
weak solution to the nonlocal problem, we have

${{{\mathcal{A}}_{\delta}}}(u_{\delta},v){+{\mathcal{Q}}(u_{\delta},v)}=(f,v)_{{\Omega}}+(g^{\delta}_{N},v)_{{{\Omega}^{N}_{{\mathcal{I}}_{\delta}}}}\,.$
(75)

The right-hand side of (75) converges to the right-hand of (73) from the
assumption on the Neumann boundary data. Using the same argument in [69,
Theorem 3.5], one can show that as ${\delta}\to 0$,

$\int_{{\Omega}}\int_{{\Omega}}{\gamma_{\delta}}({\bm{x}},{\bm{y}})(u_{\delta}({\bm{y}})-u_{\delta}({\bm{x}}))(v({\bm{y}})-v({\bm{x}})){d{\bm{y}}}{d{\bm{x}}}\to\int_{{\Omega}}\nabla
u^{\ast}\cdot\nabla v{d{\bm{x}}}\,.$

Note also that

$\begin{split}&\left|\iint_{({{\Omega}_{{\mathcal{I}}_{\delta}}}\times{\Omega})\cup({\Omega}\times{{\Omega}_{{\mathcal{I}}_{\delta}}})}{\gamma_{\delta}}({\bm{x}},{\bm{y}})(u_{\delta}({\bm{y}})-u_{\delta}({\bm{x}}))(v({\bm{y}})-v({\bm{x}})){d{\bm{y}}}{d{\bm{x}}}\right|\\\
\leq&\|u_{\delta}\|^{2}_{{V_{{\delta}}({{\widehat{{\Omega}}}_{\delta}})}}\cdot\iint_{({{\Omega}_{{\mathcal{I}}_{\delta}}}\times{\Omega})\cup({\Omega}\times{{\Omega}_{{\mathcal{I}}_{\delta}}})}{\gamma_{\delta}}({\bm{x}},{\bm{y}})(v({\bm{y}})-v({\bm{x}}))^{2}{d{\bm{y}}}{d{\bm{x}}}\\\
\leq&C\int_{{{\Omega}_{{\mathcal{I}}_{\delta}}}}|\nabla v|^{2}{d{\bm{x}}}\to
0\,,\end{split}$

as ${\delta}\to 0$. We then have the convergence of
${{{\mathcal{A}}_{\delta}}}(u_{\delta},v)$ to the integral of $\nabla
u^{\ast}\cdot\nabla v$ on ${\Omega}$. Meanwhile, by the assumption on $m$, we
have $u_{\ast}\in L^{m}({\Omega})$. Moreover, we observe that there is a
constant $C>0$ only depends on ${\Omega}$ and $m$ such that

$\displaystyle\|u_{\delta}-u^{\ast}\|_{L^{m-1}({\Omega})}^{m-1}$
$\displaystyle\leq\|u_{\delta}-u^{\ast}\|_{L^{2}({\Omega})}\|u_{\delta}-u^{\ast}\|_{L^{2(m-2)}({\Omega})}^{m-2}$
$\displaystyle\leq
C\|u_{\delta}-u^{\ast}\|_{L^{2}({\Omega})}\|u_{\delta}-u^{\ast}\|_{L^{m}({\Omega})}^{m-2}$
$\displaystyle\leq
C\|u_{\delta}-u^{\ast}\|_{L^{2}({\Omega})}(\|u_{\delta}\|_{L^{m}({\Omega})}+\|u^{\ast}\|_{L^{m}({\Omega})})^{m-2},$

where we have used $2(m-2)\leq m$ by the choice of $m$. Thus, we get the
convergence of $u_{\delta}$ to $u^{\ast}$ in $L^{m-1}({\Omega})$ (and
consequently the convergence of $q(u_{\delta})$ to $q(u^{\ast})$ in
$L^{1}({\Omega})$) from the convergence of $u_{\delta}$ to $u^{\ast}$ in
$L^{2}({\Omega})$. We thus have the convergence of
${\mathcal{Q}}(u_{\delta},v)$ to ${\mathcal{Q}}(u_{\ast},v)$. Therefore
$u^{\ast}$ is the weak solution of the local boundary value problem, and this
implies $u^{\ast}=u_{0}$. ∎

######  Remark 4.5.

An example for data satisfying (74) for ${\Omega}$ being the unit ball and
${{\Omega}^{N}_{{\mathcal{I}}_{\delta}}}={{\Omega}_{{\mathcal{I}}_{\delta}}}$
is the case of ${g_{N}^{\delta}}({\bm{x}})$ being given by
$\delta^{-1}g_{N}^{0}({\bm{x}}/|{\bm{x}}|)$ for any
${\bm{x}}\in{{\Omega}_{{\mathcal{I}}_{\delta}}}$. Similar constant extensions
along normal directions of the boundary $\partial{\Omega}$ can be constructed
for smooth ${\Omega}$. Other extensions can also be considered, see, for
example, [23, 77]. One may also examine how the convergence order depends on
the data and the solution; see, for example, [41, 42, 23]. It is no surprise
that the constant extension may not be a desirable choice in order to achieve
higher convergence order. Moreover, such a constant extension only makes sense
if the resulting nonlocal data satisfies conditions specified in the
assumption 4.1.

######  Remark 4.6.

A direct connection between the fractional Laplacian and the local Laplacian
is the case of $s\rightarrow 1^{-}$. It is known that, see e.g. [6], the
solution of integral fractional diffusion model (8) strongly converges to the
solution of the local diffusion problem in $H^{1-\epsilon}(\Omega)$. Related
discussions on Neumann boundary problems can be found in [47, 44].

######  Remark 4.7.

We may also consider the limit of Robin type problems considered in the remark
2.7 and 3.10 assuming that (26) holds. Then one may also consider the local
$\delta\to 0$ limit to the corresponding boundary value problem of the local
Laplacian with the local Robin boundary condition.

######  Remark 4.8.

The special example considered here corresponds to variational problems
associated with convex energy so that the solutions are unique. One can
consider extensions to problems associated with non-convex energy, see related
discussions in [54] using $\Gamma$-convergence.

###  4.3 Fractional limit

Here, we make more studies on treating fractional PDEs as either specialized
nonlocal models or limiting cases.

We can let $\delta=\infty$ for kernels of the types in (22). We identify some
of the spaces in this case:

$V_{\infty}({\widehat{{\Omega}}}_{\infty})=H^{s}({{\mathbb{R}}^{d}}),\quad
V_{c,\infty}({\widehat{{\Omega}}}_{\infty})={H}^{s}_{c}({{\mathbb{R}}^{d}}):=\\{u\in
H^{s}({{\mathbb{R}}^{d}})\colon u=0\
\;\forall\,{\bm{x}}\in{{{\Omega}^{D}_{{\mathcal{I}}_{\infty}}}}\\}$

######  Remark 4.9.

We consider the case that
${{\Omega}^{D}_{{\mathcal{I}}_{\infty}}}={{\Omega}_{{\mathcal{I}}_{\infty}}}$.
For $s>1/2$, then ${H}^{s}_{c}({\Omega})$ coincides with the space
$H_{0}^{s}({\Omega})$ that is the closure of $C_{0}^{\infty}({\Omega})$ with
respect to the $H^{s}({\Omega})$-norm, whereas for $s<1/2$,
${H}^{s}_{c}({\Omega})$ is isomorphic to $H^{s}({\Omega})$ (i.e., functions in
$H^{s}({\Omega})$ with zero extension to ${\Omega}_{c}$). In the critical case
$s=1/2$, ${H}^{s}_{c}({\Omega})\subsetneq H^{s}_{0}({\Omega})$. See e.g. [50,
Chapter 3] for a detailed discussion.

Let us first consider an example of a pure Dirichlet type problem. For
${{\Omega}^{D}_{{\mathcal{I}}_{\infty}}}={{\Omega}_{{\mathcal{I}}_{\infty}}}={{\Omega}_{c}}$,
the following weak formulation has been given in [3]:

Find $u({\bm{x}})\in{H}^{s}({{\mathbb{R}}^{d}})$ such that
$u({\bm{x}})=g_{D}^{\infty}({\bm{x}})$ for all ${\bm{x}}\in{{\Omega}_{c}}$ and
${\mathcal{A}}_{s}(u,v)={(f,v)_{\Omega}}{+}(g_{D}^{\infty},v)_{{{\Omega},{{\Omega}_{c}}}}\quad\text{{for
all} }v\in{H}^{s}_{c}({{\mathbb{R}}^{d}}),$ (76)

where

$\displaystyle{\mathcal{A}}_{s}(u,v)$
$\displaystyle=\frac{1}{2}\int_{({{\mathbb{R}}^{d}}\times{{\mathbb{R}}^{d}})\setminus({{\Omega}_{c}}\times{{\Omega}_{c}})}\;(u({{\bm{y}}})-u({{\bm{x}}}))(v({{\bm{y}}})-v({{\bm{x}}})){{\gamma}_{s}}{({\bm{x}},{\bm{y}})}{d{\bm{y}}d{\bm{x}}}$
$\displaystyle=\underbrace{{\frac{1}{2}}\int_{{\Omega}}\int_{{\Omega}}\;(u({{\bm{y}}})-u({{\bm{x}}}))(v({{\bm{y}}})-v({{\bm{x}}})){{\gamma}_{s}}{({\bm{x}},{\bm{y}})}{d{\bm{y}}d{\bm{x}}}}_{{\mathcal{A}}_{{\Omega},{\Omega}}(u,v)}$
$\displaystyle\quad\,+\underbrace{\int_{{\Omega}}u({{\bm{x}}})v({{\bm{x}}})\int_{{{{\Omega}_{c}}}}{{\gamma}_{s}}{({\bm{x}},{\bm{y}})}{d{\bm{y}}d{\bm{x}}}}_{{\mathcal{A}}_{{\Omega},{{\Omega}_{c}}}(u,v)}$
(77)

and

$(g_{D}^{\infty},v)_{{\Omega},{{\Omega}_{c}}}=\int_{\Omega}v({\bm{x}})\int_{{\Omega}_{c}}g_{D}^{\infty}({\bm{y}}){{\gamma}_{s}}{({\bm{x}},{\bm{y}})}{d{\bm{y}}d{\bm{x}}}.$
(78)

It is easy to see that due to the Dirichlet constraint on
${{\Omega}^{D}_{{\mathcal{I}}_{\infty}}}={{\Omega}_{{\mathcal{I}}_{\infty}}}={{\Omega}_{c}}$,
$u({\bm{x}})=g_{D}^{\infty}({\bm{x}})$ and $v({\bm{x}})=0$ for
${\bm{x}}\in{{\Omega}_{c}}$. Thus,

$\displaystyle(g_{D}^{\infty},v)_{{\Omega},{{\Omega}_{c}}}$
$\displaystyle=\int_{\Omega}v({\bm{x}})\int_{{\Omega}_{c}}g_{D}^{\infty}({\bm{y}}){{\gamma}_{s}}{({\bm{x}},{\bm{y}})}{d{\bm{y}}d{\bm{x}}}$
$\displaystyle=\int_{\Omega}v({\bm{x}})\int_{{\Omega}_{c}}u({\bm{y}}){{\gamma}_{s}}{({\bm{x}},{\bm{y}})}{d{\bm{y}}d{\bm{x}}}$

Meanwhile, assuming that Fubini’s theorem holds, then

$\displaystyle{\mathcal{A}}_{{\Omega},{{\Omega}_{c}}}(u,v)-(g_{D}^{\infty},v)_{{\Omega},{{\Omega}_{c}}}$
$\displaystyle=\int_{\Omega}\int_{{\Omega}_{c}}v({\bm{x}})(u({\bm{x}})-u({\bm{y}})){{\gamma}_{s}}{({\bm{x}},{\bm{y}})}{d{\bm{y}}d{\bm{x}}}$
$\displaystyle=\int_{\Omega}\int_{{\Omega}_{c}}(v({\bm{x}})-v({\bm{y}}))(u({\bm{x}})-u({\bm{y}})){{\gamma}_{s}}{({\bm{x}},{\bm{y}})}{d{\bm{y}}d{\bm{x}}}$
$\displaystyle=\int_{{\Omega}_{c}}\int_{\Omega}(v({\bm{x}})-v({\bm{y}}))(u({\bm{x}})-u({\bm{y}})){{\gamma}_{s}}{({\bm{x}},{\bm{y}})}{d{\bm{y}}d{\bm{x}}}$
$\displaystyle=\frac{1}{2}\int_{\Omega}\int_{{\Omega}_{c}}(v({\bm{x}})-v({\bm{y}}))(u({\bm{x}})-u({\bm{y}})){{\gamma}_{s}}{({\bm{x}},{\bm{y}})}{d{\bm{y}}d{\bm{x}}}$
$\displaystyle\quad+\frac{1}{2}\int_{{\Omega}_{c}}\int_{\Omega}(v({\bm{x}})-v({\bm{y}}))(u({\bm{x}})-u({\bm{y}})){{\gamma}_{s}}{({\bm{x}},{\bm{y}})}{d{\bm{y}}d{\bm{x}}}$

Thus, (76) is equivalent to that defined (61) for $\delta=\infty$ with
${\mathcal{Q}}=0$.

######  Remark 4.10.

One may also enforce the volume constraints via Lagrange multipliers, see e.g.
[2].

One can derive some rigorous results on the limit of (12) as $\delta\to\infty$
for kernels satisfying (22), particularly for the case of pure Dirichlet
condition. For example, one can extend the result given [75] for the case of
${g_{D}^{\delta}}\equiv 0$ and $g_{D}^{\infty}\equiv 0$ (homogeneous Dirichlet
problem) to inhomogeneous data.

######  Theorem 4.11.

For kernels satisfying (22), let $u_{\delta}$ and $u_{\infty}$ be the weak
solutions to the nonlocal and fractional Dirichlet boundary value problems
associated with an element $w\in H^{s}({{\mathbb{R}}^{d}})$ and
${g_{D}^{\delta}}=w\mid_{{{\Omega}_{{\mathcal{I}}_{\delta}}}}$ and
$g_{D}^{\infty}=w\mid_{{{\Omega}_{c}}}$ respectively. Assume that
$\|{\widehat{f}}\|_{V^{\star}_{{\delta}}({{\widehat{{\Omega}}}_{\delta}})}$ is
bounded uniformly in ${\delta}$, then we have

$\|u_{\delta}-u_{\infty}\|_{H^{s}({\Omega})}\to 0\,,$

as ${\delta}\to\infty$.

Indeed, one can get the above conclusion by considering the corresponding
problems for $u_{\delta}-w$ and $u_{\infty}-w$ with homogeneous Dirichlet data
and noticing that ${{\mathcal{A}}_{\delta}}(w,v)\to{\mathcal{A}}_{s}(w,v)$ as
${\delta}\to\infty$ for any $v\in H^{s}_{c}({{\mathbb{R}}^{d}})$. Then the
convergence of $u_{\delta}\to u_{\infty}$ can be derived from a similar
convergence result for the homogeneous Dirichlet problem, see [75] for more
discussions on the latter.

######  Remark 4.12.

In the more general case, further studies are needed to characterize the
solution spaces and the weak form as ${\delta}\to\infty$ for kernels
satisfying (22). For example, in [26], some weighted norm based on the Neumann
data $g$ is introduced, which is involved in the definition of weak solution.

##  5 Numerical experiments and discussion

In this section, we shall present some numerical examples to show the
behaviours of the solution to the nonlocal model (12) with inhomogeneous
boundary conditions. Since the Dirichlet boundary conditions have been
intensively studied in existing works, we focus on the cases for the pure
Neumann or mixed type boundary conditions here.

We consider a one-dimensional nonlocal problem in the unit interval
$\Omega=(0,1)$ with boundary region
${{\Omega}_{{\mathcal{I}}_{\delta}}}=(-\delta,0)\cup(1,1+\delta)$. Throughout
the section, we choose the kernel functions in the fractional type:

$\gamma_{\delta}(x,y)=C_{s,\delta}|x-y|^{-1-2s}\chi_{B_{\delta}(0)}(x-y)$

Here the normalization constant $C_{s,\delta}(\delta)$ is defined by

$C_{s,\delta}=\begin{cases}(2-2s)\delta^{-2+2s},\quad&\text{for}~{}~{}\delta\leq\delta_{s},\\\
\displaystyle\frac{2^{2s}s\Gamma(s+{{1}/{2}})}{\pi^{\frac{1}{2}}\Gamma(1-s)},\quad&\text{for}~{}~{}\delta\geq\delta_{s},\end{cases}\quad\text{with}\quad\delta_{s}=\Big{(}\displaystyle\frac{(2-2s)\pi^{\frac{1}{2}}\Gamma(1-s)}{2^{2s}s\Gamma(s+1/2)}\Big{)}^{\frac{1}{2-2s}}.$

In our computation, we use the standard Galerkin finite element method with
uniform mesh size $h=1/M$. Without loss of generality, we assume that
$h\ll\delta$ and $r=\delta/h$ is an integer. Define the grid points
$\\{x_{r}\\}_{i=-r}^{M+r}$ such that $x_{r}=rh$, we then define $X_{h}$ to be
the set of functions in ${V_{c,{\delta}}({{\widehat{{\Omega}}}_{\delta}})}$
(see the definition in (51)) which are linear when restricted to the
subintervals $[x_{i},x_{i+1}]$, $i=-r,\ldots,M+r$. We fix a sufficiently small
mesh size $h=10^{-4}$.

Example 1. Local limit: pure Neumann boundary conditions. To begin with, we
examine the nonlocal model (12) with the inhomogeneous Neumann boundary
conditions. We define the source term $f(x)=x(1-x)$ for all $x\in\Omega$ and
the Neumann type volumetric constraint

$g_{N}^{\delta}(x)=\begin{cases}-\displaystyle\frac{1}{6\delta},\quad&x\in(-\delta,0),\\\
0,\quad&x\in(1,1+\delta).\end{cases}$

Note that the source term $f$ and boundary data $g_{N}^{\delta}$ satisfy the
compatibility condition (67), which implies the existence of the weak
solution. We shall test two types of Neumann Boundary conditions that
correspond to the bilinear forms (62) and (69) respectively, and thus implying
implicitly the use of the operators ${{\mathcal{N}}_{\delta}}$ and
${\mathcal{N}}_{\delta,{{\widehat{{\Omega}}}_{\delta}}}$ respectively, where

${{\mathcal{N}}_{\delta}}u(x):=-\int_{\Omega}(u(y)-u(x))\gamma_{\delta}(|x-y|)\,\mbox{d}y,\quad\forall\,x\in{{\Omega}_{{\mathcal{I}}_{\delta}}}.$
(79)

and

${\mathcal{N}}_{\delta,{{\widehat{{\Omega}}}_{\delta}}}u(x):=-\int_{{\widehat{{\Omega}}}_{\delta}}(u(y)-u(x))\gamma_{\delta}(|x-y|)\,\mbox{d}y,\quad\forall\,x\in{{\Omega}_{{\mathcal{I}}_{\delta}}}.$
(80)

For both cases, the local limit is the steady diffusion problem

$-u^{\prime\prime}(x)=f(x)~{}~{}\text{for}~{}~{}x\in\Omega,\qquad\text{with}~{}~{}u^{\prime}(0)=1/6~{}~{}\text{and}~{}~{}u^{\prime}(1)=0.$
(81)

With the nonlocal Neumann boundary condition
${{\mathcal{N}}_{\delta}}u=g_{N}^{\delta}$, the finite element scheme for the
nonlocal problem reads: find $u_{h}\in X_{h}$ such that for all $v_{h}\in
X_{h}$,

$\begin{split}{{{\mathcal{A}}_{\delta}}(u_{h},v_{h})=}\frac{1}{2}\int_{{\widetilde{{\Omega}}_{\delta}}}(v_{h}(y)-v_{h}(x))&(u_{h}(y)-u_{h}(x))\gamma_{\delta}(x-y)\,dy\,dx\\\
&=\int_{\Omega}f(x)v_{h}(x)\,dx+\int_{\Omega_{\mathcal{I}_{\delta}}}g_{N}^{\delta}v_{h}(x)\,dx.\end{split}$
(82)

Meanwhile, with the second kind of Neumann boundary condition
${\mathcal{N}}_{\delta,{{\widehat{{\Omega}}}_{\delta}}}u=g_{N}^{\delta}$, the
corresponding finite element approximation reads: find $u_{h}\in X_{h}$ such
that for all $v_{h}\in X_{h}$,

$\begin{split}{{\mathcal{A}}_{\delta,{{\widehat{{\Omega}}}_{\delta}}}(u_{h},v_{h})=}\frac{1}{2}\int_{{{\widehat{{\Omega}}}_{\delta}}\times{{\widehat{{\Omega}}}_{\delta}}}&(v_{h}(y)-v_{h}(x))(u_{h}(y)-u_{h}(x))\gamma_{\delta}(x-y)\,dy\,dx\\\
&=\int_{\Omega}f(x)v_{h}(x)\,dx+\int_{\Omega_{\mathcal{I}_{\delta}}}g_{N}^{\delta}v_{h}(x)\,dx.\end{split}$
(83)

Both finite dimensional problems (82) and (83) are well-posed by the
Lax–Milgram theorem and the nonlocal version of Poincaré inequality in Lemma
2.

Figure 5.1 presents the numerical solutions in $(0,1)$ with various $\delta$
and $s$, while Figure 5.2 zooms in on the solution profile near the boundary
$x=0$. In the case that $s=-1$, the solution is nonsmooth and we observe
discontinuity at $x=0$, while the solution is continuous in
${{\widehat{{\Omega}}}_{\delta}}$ in cases that $s=0.25$ and $0.75$. As
$\delta\rightarrow 0$, the numerical results clearly indicate that the
solution to the nonlocal diffusion problem converges to $u_{0}$, the solution
to the local problem. This fully supports our theoretical finding in Theorem
4.4. Meanwhile, with the relatively large nonlocal horizon $\delta$, our
empirical experiments also show that the solution of the boundary operator
${\mathcal{N}}_{\delta,{{\widehat{{\Omega}}}_{\delta}}}$ is closer to the
local solution $u_{0}$. Finally, in Table 1, we present the convergence rate
as $\delta\rightarrow 0$ for both cases. These interesting phenomena warrant
further investigation in our future studies.

$s=0.75$ | $s=0.25$ | $s=-1$
---|---|---
Fig. 5.1: Example 1. Solutions to the nonlocal diffusion problem with inhomogeneous Neumann type boundary conditions, with operators ${{\mathcal{N}}_{\delta}}$ (upper) and ${\mathcal{N}}_{\delta,{{\widehat{{\Omega}}}_{\delta}}}$ (lower). $s=0.75$ | $s=0.25$ | $s=-1$
---|---|---
Fig. 5.2: Example 1. Profiles of $u_{\delta}(\delta y)$, $y\in(-1/2,1)$, with various horizon parameters $\delta$ and fractional powers $s$, with operators ${{\mathcal{N}}_{\delta}}$ (upper) and ${\mathcal{N}}_{\delta,{{\widehat{{\Omega}}}_{\delta}}}$ (lower). Table 1: Example 1. $\|u_{\delta}-u_{0}\|_{L^{2}(\Omega)}$ with $s=0.75$, $0.25$ and $-1$. | $s\backslash\delta$ | 0.08 | 0.04 | 0.02 | 0.01 | 0.005 | Rate ($\delta$)
---|---|---|---|---|---|---|---
| $-1$ | 9.33E-3 | 3.02E-3 | 1.14E-3 | 4.35E-4 | 1.97E-4 | 1.27
$\mathcal{N}_{\delta}$ | $0.25$ | 2.15E-2 | 6.29E-3 | 1.92E-3 | 6.54E-4 | 2.48E-4 | 1.47
| $0.75$ | 5.97E-2 | 1.63E-2 | 4.44E-3 | 1.27E-3 | 3.95E-4 | 1.75
| $-1$ | 2.71E-3 | 1.33E-3 | 6.63E-4 | 3.27E-4 | 1.60E-4 | 1.02
${\mathcal{N}}_{\delta,{{\widehat{{\Omega}}}_{\delta}}}$ | $0.25$ | 2.65E-3 | 1.32E-3 | 6.60E-4 | 3.26E-4 | 1.60E-4 | 1.01
| $0.75$ | 2.62E-3 | 1.31E-3 | 6.57E-4 | 3.26E-4 | 1.60E-4 | 1.01

Example 2. Local limit: mixed boundary conditions. In this example, we
consider zero source term, i.e. $f(x)=0$, and the following two types of
boundary conditions:

$\begin{split}&\mathcal{N}_{\delta}u(x)=-\frac{1}{3\delta}~{}~{}\text{for}~{}~{}x\in(-\delta,-\delta/2),\quad
u(x)=1~{}~{}\text{for}~{}~{}x\in[-\delta/2,0),\\\
&u(x)=1~{}~{}\text{for}~{}~{}x\in(1,1+\delta),\qquad\textbf{Example
2(a)},\end{split}$

and

$\begin{split}&u(x)=1~{}~{}\text{for}~{}~{}x\in(-\delta,-\delta/2),\quad\mathcal{N}_{\delta}u(x)=-\frac{1}{3\delta}~{}~{}\text{for}~{}~{}x\in[-\delta/2,0),\\\
&u(x)=1~{}~{}\text{for}~{}~{}x\in(1,1+\delta),\qquad\textbf{Example
2(b)}.\end{split}$

Noting that the main difference between Examples 2(a) and 2(b) is the order of
Neumann and Dirichlet boundary conditions in the interval $(-\delta,0)$. This
leads to the case that a part of the data (Neumann data for the former, and
Dirichlet data for the latter respectively) is defined in the interval
disconnected from $(0,1)$. It is interesting to observe that, in Figure 5.3,
both solutions converge to the same local limit, i.e., the solution to the
steady diffusion problem: $-u^{\prime\prime}(x)=0$ in $\Omega=(0,1)$ with
Dirichlet boundary conditions $u(0)=u(1)=1$. The solution profiles near $x=0$,
i.e. $u_{\delta}(\delta y)$ with $y\in(-1,2)$, are plotted in Figure 5.4. We
observe the discontinuity at $x=-\delta/2$, which might be due to the
incompatibility between the nonlocal Neumann and Dirichlet boundary
conditions. In Table 2, we present the difference between $u_{\delta}$ and
$u_{0}$ in the $L^{2}$ sense. Our numerical experiments show that the
convergence of Example 2 (b) is slower than that of Example 2 (a). These
observations deserve further theoretical analysis in the future.

$s=0.75$ | $s=0.25$ | $s=-1$
---|---|---
Fig. 5.3: Example 2 (a) (upper) and 2 (b) (lower). Solutions to the nonlocal diffusion problem with various horizon parameters $\delta$ and fractional powers $s$.  $s=0.75$ | $s=0.25$ | $s=-1$
---|---|---
Fig. 5.4: Example 2(a)(upper) and 2(b)(lower). Profiles of $u_{\delta}(\delta y)$, $y\in(-1,2)$, with various horizon parameters $\delta$ and fractional powers $s$. Table 2: Example 2 (a) and (b). $\|u_{\delta}-u_{0}\|_{L^{2}(\Omega)}$ with $s=0.75$, $0.25$ and $-1$. | $s\backslash\delta$ | 0.16 | 0.08 | 0.04 | 0.02 | 0.01 | Rate ($\delta$)
---|---|---|---|---|---|---|---
| $-1$ | 1.67E-2 | 8.04E-3 | 3.95E-3 | 1.96E-3 | 9.84E-4 | 1.02
(a) | $0.25$ | 5.11E-3 | 2.53E-3 | 1.27E-3 | 6.40E-4 | 3.27E-5 | 0.99
| $0.75$ | 2.32E-3 | 1.17E-3 | 5.92E-4 | 3.01E-4 | 1.55E-4 | 0.98
| $-1$ | 3.33E-2 | 1.93E-2 | 1.06E-2 | 5.63E-3 | 2.96E-3 | 0.87
(b) | $0.25$ | 3.90E-2 | 2.38E-2 | 1.35E-2 | 7.31E-3 | 3.90E-3 | 0.83
| $0.75$ | 5.87E-2 | 4.15E-2 | 2.63E-2 | 1.54E-2 | 8.56E-3 | 0.69

Example 3. Fractional limit. In this example, we shall numerically check the
limit of nonlocal diffusion models as $\delta\rightarrow\infty$. To this end,
we consider the boundary value problem of steady fractional diffusion model

$\begin{cases}-{{\mathcal{L}}_{\infty}^{s}}u_{\infty}(x)+u_{\infty}=f(x),&\quad\forall\,x\in{\Omega}{,}\\\
\mathcal{N}_{\infty}^{s}u_{\infty}(x)=g(x),&\quad\forall\,x\in{{\Omega}_{c}},\end{cases}$
(84)

where the fractional operator ${{\mathcal{L}}_{\infty}^{s}}$ and the Neumann
operator $\mathcal{N}_{\infty}^{s}$ are defined in (6) and (10), respectively.
Since the closed form of the fractional model is unavailable, we assume that
the solution is a Gaussian function:

$u_{\infty}(x)=\exp\big{(}-(x-{{1}/{2}})^{2}\big{)}\qquad\forall~{}~{}x\in\mathbb{R}.$
(85)

Then it is easy to compute the source term $f$ and the Neumann boundary data
$g$ numerically, since $u_{\infty}(x)$ decays double exponentially as
$|x|\rightarrow\infty$. In our computation, we evaluate the numerical solution
of the following boundary value problem of the steady nonlocal diffusion
problem with a finite horizon:

$\begin{cases}-\mathcal{L}_{\delta}u_{\delta}(x)+u_{\delta}=f(x),&\quad\forall\,x\in{\Omega}{,}\\\
\mathcal{N}_{\delta}u_{\delta}(x)=g(x),&\quad\forall\,x\in\Omega_{\mathcal{I}_{\delta}},\end{cases}$
(86)

Note that, by adding lower order terms $u_{\infty}$ and $u_{\delta}$ in (84)
and (86) respectively, we will not need to be concerned with data
incompatibility for the corresponding fractional and nonlocal problems.

In Figure 5.5, we plot the numerical solutions $u_{\delta}$ with various
$\delta$ and $s$. The numerical results clearly show the convergence of
$u_{\delta}$ to $u_{\infty}$ as $\delta\rightarrow\infty$. Besides, for
smaller $s$, we observe a slower convergence, which might be due to the slower
decay of the kernel function in the case of smaller $s$.

Similar to Example 1, we also consider the boundary value problem with another
type of Neumann boundary condition:

$\begin{cases}-{{\mathcal{L}}_{\infty}^{s}}u_{\infty}(x)+u_{\infty}=f(x),&\quad\forall\,x\in{\Omega}{,}\\\
\widetilde{\mathcal{N}}_{\infty}^{s}u_{\infty}(x)=\widetilde{g}(x),&\quad\forall\,x\in{{\Omega}_{c}},\end{cases}$
(87)

where the fractional Neumann operator is defined by

$\widetilde{\mathcal{N}}_{\infty}^{s}u(x)=-C_{s,\infty}\int_{\mathbb{R}}\frac{u(y)-u(x)}{|x-y|^{1+2s}}\,dy,\quad\quad\forall\,x\in{{{\Omega}_{c}}}.$
(88)

Let the solution to (87) be of the form (85). Then we are able to compute the
new Neumann boundary data
$\widetilde{g}(x)=\widetilde{\mathcal{N}}_{\infty}^{s}u_{\infty}(x)$. The
corresponding nonlocal model reads

$\begin{cases}-\mathcal{L}_{\delta}u_{\delta}(x)+u_{\delta}=f(x),&\quad\forall\,x\in{\Omega}{,}\\\
{\mathcal{N}}_{\delta,{{\widehat{{\Omega}}}_{\delta}}}u_{\delta}(x)=\widetilde{g}(x),&\quad\forall\,x\in\Omega_{\mathcal{I}_{\delta}},\end{cases}$
(89)

Again, our numerical results indicate that the functions $u_{\delta}$ converge
to $u_{\infty}$ as $\delta\rightarrow\infty$, and the convergence rate
deteriorates as $s\rightarrow 0$, see Figure 5.5. For both cases, we observe
that the convergence rate is around $O(\delta^{-2s})$(see Table 3), which
still awaits theoretical justification.

$s=0.75$ | $s=0.5$ | $s=0.25$
---|---|---
Fig. 5.5: Example 3. Solutions to the nonlocal diffusion problem with inhomogeneous Neumann type boundary conditions, with operators ${{\mathcal{N}}_{\delta}}$ (upper) and ${\mathcal{N}}_{\delta,{{\widehat{{\Omega}}}_{\delta}}}$ (lower). Table 3: Example 3. $\|u_{\delta}-u_{\infty}\|_{L^{2}(\Omega)}$ with $s=0.25$, $0.5$ and $0.75$. |  $s\backslash$ $\delta$ | 10 | 20 | 40 | 80 | 160 | Rate ($\delta$)
---|---|---|---|---|---|---|---
| 0.25 | 2.25E-1 | 1.61E-1 | 1.15E-1 | 8.12E-2 | 5.75E-2 | 0.49
$\mathcal{N}_{\delta}$ | 0.50 | 5.55E-2 | 2.84E-2 | 1.44E-2 | 7.22E-3 | 3.62E-3 | 0.98
| 0.75 | 1.07E-2 | 3.93E-3 | 1.41E-3 | 5.03E-4 | 1.78E-4 | 1.48
| 0.25 | 4.40E-1 | 3.15E-1 | 2.24E-1 | 1.59E-1 | 1.12E-1 | 0.49
${\mathcal{N}}_{\delta,{{\widehat{{\Omega}}}_{\delta}}}$ | 0.50 | 1.01E-1 | 5.56E-2 | 2.81E-2 | 1.41E-2 | 7.08E-3 | 0.99
| 0.75 | 2.11E-2 | 7.69E-3 | 2.76E-3 | 9.84E-3 | 3.48E-3 | 1.48

##  6 Conclusion

The discussion here offers a glimpse into some of the recent studies of
nonlocal models with a finite range of interactions defined on a bounded
domain involving various types of nonlocal boundary conditions. In particular,
we provide a consistent presentation of such models to classical PDEs, seen as
the local limits, and fractional PDEs, seen as the limit with an infinite
range of interactions. Obviously, there are many more known results in the
literature and more interesting and open questions to be considered. For
example, while the associated nonlocal boundary conditions are illustrated for
problems defined in a bounded domain, the inhomogeneous data are only
prescribed in abstract function spaces. It is interesting to give more precise
characterizations of these spaces by applying recently developed trace and
extension theorems for the nonlocal function spaces [38, 43]. Also, while we
stress the consistent local and fractional limits, one can also further
examine how the rates of convergence depend on the model and data in these
limits [41, 42, 23]. Moreover, the formulations are only presented in the
context of time-independent scalar equations, one can consider the extensions
to nonlinear systems and evolution equations, particularly those connected to
peridynamics [64, 40, 36, 28]. Another interesting topic, as previously
mentioned in the discussion, is on the connection of the nonlocal problems to
the stochastic process and their limits, from the local diffusion equation and
Brownian motion to fractional diffusion and Lévy jump processes [26, 39].

In addition, one may consider the numerical analysis of approximation methods
and discuss, among many practical issues, the issue of asymptotic compatible
approximation [72]. Further numerical experiments can also be carried out to
provide more comparisons of the nonlocal models and their various limits.

## References

  * [1] N. Abatangelo and L. Dupaigne, Nonhomogeneous boundary conditions for the spectral fractional Laplacian, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2017), pp. 439–467.
  * [2] G. Acosta, J.-P. Borthagaray, and N. Heuer, Finite element approximations of the nonhomogeneous fractional Dirichlet problem, IMA J. Numer. Anal., 39 (2019), pp. 1471–1501.
  * [3] M. Ainsworth and C. Glusa, Towards an efficient finite element method for the integral fractional Laplacian on polygonal domains, in Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan, Springer, 2018, pp. 17–57.
  * [4] F. Andreu, J. Mazón, J. Rossi, and J. Toledo, Nonlocal Diffusion Problems, vol. 165 of Mathematical Surveys and Monographs, American Mathematical Society, 2010.
  * [5] H. Antil, J. Pfefferer, and S. Rogovs, Fractional operators with inhomogeneous boundary conditions: analysis, control, and discretization, Commun. Math. Sci., 16 (2018), pp. 1395–1426.
  * [6] U. Biccari and V. Hernández-Santamaría, The Poisson equation from non-local to local, Electron. J. Differential Equations, (2018), pp. Paper No. 145, 13.
  * [7] K. Bogdan, K. Burdzy, and Z.-Q. Chen, Censored stable processes, Probab. Theory Related Fields, 127 (2003), pp. 89–152.
  * [8] A. Bonito, J. P. Borthagaray, R. H. Nochetto, E. Otárola, and A. J. Salgado, Numerical methods for fractional diffusion, Computing and Visualization in Science, 19 (2018), pp. 19–46.
  * [9] J. Bourgain, H. Brezis, and P. Mironescu, Another look at Sobolev spaces, IOS Press, 2001, pp. 439–455.
  * [10] C. Bucur and E. Valdinoci, Nonlocal diffusion and applications, vol. 1, Springer, 2016.
  * [11] D. Burago, S. Ivanov, and Y. Kurylev, A graph discretization of the Laplace-Beltrami operator, Journal of Spectral Theory, 4 (2014), pp. 675–714.
  * [12] O. Burkovska and M. Gunzburger, Regularity and approximation analyses of nonlocal variational equality and inequality problems, J. Math. Anal. Appl., 478 (2019), pp. 1027–1048.
  * [13] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Communications in partial differential equations, 32 (2007), pp. 1245–1260.
  * [14] Z.-Q. Chen and P. Kim, Green function estimate for censored stable processes, Probab. Theory Related Fields, 124 (2002), pp. 595–610.
  * [15] R. R. Coifman and S. Lafon, Diffusion maps, Applied and Computational Harmonic Analysis, 21 (2006), pp. 5 – 30.  Special Issue: Diffusion Maps and Wavelets.
  * [16] R. R. Coifman, S. Lafon, A. B. Lee, M. Maggioni, F. Warner, and S. Zucker, Geometric diffusions as a tool for harmonic analysis and structure definition of data: Diffusion maps, in Proceedings of the National Academy of Sciences, 2005, pp. 7426–7431.
  * [17] C. Cortazar, M. Elgueta, J. Rossi, and N. Wolanski, How to approximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems, Archive for Rational Mechanics and Analysis, 187 (2008), pp. 137–156.
  * [18] G.-H. Cottet and P. D. Koumoutsakos, Vortex methods: theory and practice, Cambridge university press, 2000.
  * [19] N. Cusimano, F. del Teso, L. Gerardo-Giorda, and G. Pagnini, Discretizations of the spectral fractional Laplacian on general domains with Dirichlet, Neumann, and Robin boundary conditions, SIAM J. Numer. Anal., 56 (2018), pp. 1243–1272.
  * [20] M. D’Elia, Q. Du, C. Glusa, M. Gunzburger, X. Tian, and Z. Zhou, Numerical methods for nonlocal and fractional models, Acta Numerica, 29 (2020), pp. 1–124.
  * [21] M. D’Elia, Q. Du, M. Gunzburger, and R. B. Lehoucq, Nonlocal convection-diffusion problems on bounded domains and finite-range jump processes, Comput. Meth. in Appl. Math., 17 (2017), pp. 707–722.
  * [22] M. D’Elia and M. Gunzburger, The fractional Laplacian operator on bounded domains as a special case of the nonlocal diffusion operator, Comput. Math. Appl., 66 (2013), pp. 1245–1260.
  * [23] M. D’Elia, X. Tian, and Y. Yu, A physically consistent, flexible, and efficient strategy to convert local boundary conditions into nonlocal volume constraints, SIAM Journal on Scientific Computing, 42 (2020), pp. A1935–A1949.
  * [24] W. Deng, B. Li, W. Tian, and P. Zhang, Boundary problems for the fractional and tempered fractional operators, Multiscale Modeling & Simulation, 16 (2018), pp. 125–149.
  * [25] E. Di Nezza, G. Palatucci, and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), pp. 521–573.
  * [26] S. Dipierro, X. Ros-Oton, and E. Valdinoci, Nonlocal problems with Neumann boundary conditions, Rev. Mat. Iberoam., 33 (2017), pp. 377–416.
  * [27] Q. Du, Local limits and asymptotically compatible discretizations, in Handbook of peridynamic modeling, Adv. Appl. Math., CRC Press, Boca Raton, FL, 2017, pp. 87–108.
  * [28] Q. Du, Nonlocal Modeling, Analysis, and Computation, vol. 94 of CBMS-NSF Conference Series in Applied Mathematics, SIAM, 2019.
  * [29] Q. Du, B. Engquist, and X. Tian, Multiscale modeling, homogenization and nonlocal effects: mathematical and computational issues, in Contemporary Mathematics, Celebrating 75th Anniversary of Mathematics of Computation, vol. 754, AMS, 2020, pp. 115–140.
  * [30] Q. Du, M. Gunzburger, R. Lehoucq, and K. Zhou, Analysis and approximation of nonlocal diffusion problems with volume constraints, SIAM Review, 54 (2012), pp. 667–696.
  * [31] , A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws, Math. Mod. Meth. Appl. Sci., 23 (2013), pp. 493–540.
  * [32] Q. Du, Z. Huang, and R. Lehoucq, Nonlocal convection-diffusion volume constrained problems and jump processes, Disc. Cont. Dyn. Syst. B, 19 (2014), pp. 373–389.
  * [33] Q. Du, L. Ju, X. Li, and Z. Qiao, Maximum bound principle for a class of semilinear parabolic equations and exponential time differencing schemes, SIAM Review, 63 (2021), pp. 317–359.
  * [34] Q. Du, T. Mengesha, and X. Tian, $L^{p}$ compactness criteria with an application to variational convergence of some nonlocal energy functionalss, Preprint, (2022).
  * [35] Q. Du and Z. Shi, A nonlocal stokes system with volume constraints, Numerical Mathematics, Theory, Methods and Applications, under review (2022).
  * [36] Q. Du, Y. Tao, and X. Tian, A peridynamic model of fracture mechanics with bond-breaking, Journal of Elasticity, 132 (2018), pp. 197–218.
  * [37] Q. Du and X. Tian, Mathematics of smoothed particle hydrodynamics via a nonlocal stokes equation, Foundations of Computational Mathematics, 20 (2020), pp. 801–826.
  * [38] Q. Du, X. Tian, C. Wright, and Y. Yu, Nonlocal trace spaces and extension results for nonlocal calculus, J. Functional Analysis, to appear, arXiv preprint arXiv:2107.00177 (2022).
  * [39] Q. Du, L. Toniazzi, and Z. Zhou, Stochastic representation of solution to nonlocal-in-time diffusion, Stochastic Processes and their Applications, 130 (2020), pp. 2058–2085.
  * [40] Q. Du, J. Yang, and Z. Zhou, Analysis of a nonlocal-in-time parabolic equation, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), pp. 339–368.
  * [41] Q. Du and X. Yin, A conforming DG method for linear nonlocal models with integrable kernels, J. Scientific Computing, 80 (2019), pp. 1913–1935.
  * [42] Q. Du, J. Zhang, and C. Zheng, On uniform second order nonlocal approximations to linear two-point boundary value problems, Communications in Mathematical Sciences, 17 (2019), pp. 1737–1755.
  * [43] B. Dyda and M. Kassmann, Function spaces and extension results for nonlocal dirichlet problems, Journal of Functional Analysis, 277 (2019), p. 108134.
  * [44] G. Foghem and M. Kassmann, A general framework for nonlocal neumann problems, arXiv preprint arXiv:2204.06793, (2022).
  * [45] M. Foss, P. Radu, and Y. Yu, Convergence analysis and numerical studies for linearly elastic peridynamics with dirichlet-type boundary conditions, arXiv preprint arXiv:2106.13878, (2021).
  * [46] G. Gilboa and S. Osher, Nonlocal operators with applications to image processing, Multi. Model. Simul., 7 (2008), pp. 1005–1028.
  * [47] G. F. F. Gounoue, $L^{2}$-Theory for nonlocal operators on domains, PhD thesis, Bielefeld university, 2020.
  * [48] Z. Huang, Nonlocal Models with Convection Effects, PhD thesis, Pennsylvania State University, 2015.
  * [49] Z. Li, Z. Shi, and J. Sun, Point integral method for elliptic equations with variable coefficients on point cloud, Communications in Computational Physics, 26 (2019), pp. 506–530.
  * [50] W. McLean, Strongly elliptic systems and boundary integral equations, Cambridge University Press, Cambridge, 2000.
  * [51] T. Mengesha, Nonlocal korn-type characterization of sobolev vector fields, Communications in Contemporary Mathematics, 14 (2012), p. 1250028.
  * [52] T. Mengesha and Q. Du, Analysis of a scalar nonlocal peridynamic model with aign changing kernel, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), pp. 1415–1437.
  * [53] , The bond-based peridynamic system with dirichlet-type volume constraint, Proceedings of the Royal Society of Edinburgh A: Mathematics, 144 (2014), pp. 161–186.
  * [54] , On the variational limit of a class of nonlocal functionals related to peridynamics, Nonlinearity, 28 (2015), pp. 3999–4035.
  * [55] , Characterization of function spaces of vector fields and an application in nonlinear peridynamics, Nonlinear Anal., 140 (2016), pp. 82–111.
  * [56] R. Metzler and J. Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep., 339 (2000), p. 77.
  * [57] R. Metzler and J. Klafter, The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics, Journal of Physics A: Mathematical and General, 37 (2004), p. R161.
  * [58] J. Monaghan, Smoothed particle hydrodynamics, Reports on progress in physics, 68 (2005), p. 1703.
  * [59] R. Nochetto, E. Otárola, and A. J. Salgado, A PDE approach to fractional diffusion in general domains: a priori error analysis, Found. Comput. Math., 15 (2015), pp. 733–791.
  * [60] G. Pang, M. D’Elia, M. Parks, and G. E. Karniadakis, nPINNs: nonlocal physics-informed neural networks for a parametrized nonlocal universal Laplacian operator. algorithms and applications, Journal of Computational Physics, 422 (2020), p. 109760.
  * [61] A. C. Ponce, An estimate in the spirit of Poincaré’s inequality, J. Eur. Math. Soc, 6 (2004), pp. 1–15.
  * [62] R. Servadei and E. Valdinoci, On the spectrum of two different fractional operators, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), pp. 831–855.
  * [63] Z. Shi and J. Sun, Convergence of the point integral method for Laplace–Beltrami equation on point cloud, Research in the Mathematical Sciences, 4 (2017), pp. 1–39.
  * [64] S. Silling, Reformulation of elasticity theory for discontinuities and long-range forces, Journal of the Mechanics and Physics of Solids, 48 (2000), pp. 175–209.
  * [65] , Introduction to Peridynamics, in Handbook of Peridynamic Modeling, Adv. Appl. Math., CRC Press, Boca Raton, FL, 2017, pp. 25–59.
  * [66] I. Sokolov, J. Klafter, and A. Blumen, Fractional kinetics, Physics Today, 55 (2002), pp. 48–54.
  * [67] P. R. Stinga, Fractional powers of second order partial differential operators: extension problem and regularity theory, PhD thesis, UAM, 2010.
  * [68] P. R. Stinga, User’s guide to the fractional Laplacian and the method of semigroups, in Handbook of fractional calculus with applications. Vol. 2, De Gruyter, Berlin, 2019, pp. 235–265.
  * [69] Y. Tao, X. Tian, and Q. Du, Nonlocal diffusion and peridynamic models with Neumann type constraints and their numerical approximations, Appl. Math. Comput., 305 (2017), pp. 282–298.
  * [70] H. Tian, L. Ju, and Q. Du, Nonlocal convection-diffusion problems and finite element approximations, Comput. Methods Appl. Mech. Engrg., 289 (2015), pp. 60–78.
  * [71] , A conservative nonlocal convection-diffusion model and asymptotically compatible finite difference discretization, Comput. Methods Appl. Mech. Engrg., 320 (2017), pp. 46–67.
  * [72] X. Tian and Q. Du, Asymptotically compatible schemes and applications to robust discretization of nonlocal models, SIAM J. Numer. Anal., 52 (2014), pp. 1641–1665.
  * [73] , Nonconforming discontinuous Galerkin methods for nonlocal variational problems, SIAM J. Numer. Anal., 53 (2015), pp. 762–781.
  * [74] , Asymptotically compatible schemes for robust discretization of parametrized problems with applications to nonlocal models, SIAM Review, 62 (2020).
  * [75] X. Tian, Q. Du, and M. Gunzburger, Asymptotically compatible schemes for the approximation of fractional Laplacian and related nonlocal diffusion problems on bounded domains, Adv. Comput. Math., 42 (2016), pp. 1363–1380.
  * [76] E. Valdinoci, From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl. SeMA, 49 (2009), pp. 33–44.
  * [77] H. You, X. Lu, N. Task, and Y. Yu, An asymptotically compatible approach for neumann-type boundary condition on nonlocal problems, ESAIM: Mathematical Modelling and Numerical Analysis, 54 (2020), pp. 1373–1413.
  * [78] Y. Zhang and Z. Shi, A second-order nonlocal approximation for surface Poisson model with dirichlet boundary, arXiv preprint arXiv:2101.01016, (2021).