Maha Daoud
email: mahaadaoud@gmail.com

Haj El

Laamri

FRACTIONAL LAPLACIANS : A SHORT SURVEY

Keywords: 2020 Mathematics Subject Classification. nonlocal operator, fractional laplacian, regional laplacian, restricted Laplacian, spectral Laplacian, peridynamic Laplacian Fractional Laplacian, nonlocal operators, regional fractional Laplacian, spectral fractional Laplacian, fractional Poisson equations

This paper describes the state of the art and gives a survey of the wide literature published in the last years on the fractional Laplacian. We will first recall some definitions of this operator in R N and its main properties. Then, we will introduce the four main operators often used in the case of a bounded domain ; and we will give several simple and significant examples to highlight the difference between these four operators. Also we give a rather long list of references : it is certainly not exhaustive but hopefully rich enough to track most connected results. We hope that this short survey will be useful for young researchers of all ages who wish to have a quick idea of the fractional Laplacian(s).

1. Introduction. The theory of fractional powers of operators on Banach spaces is a classical topic in mathematical analysis and Probability. It was developed over the last century, we refer the interested reader for instance to [START_REF] Balakrishnan | Fractional powers of closed operators and the semigroups generated by them[END_REF][START_REF] Bochner | Diffusion equation and stochastic processes[END_REF][START_REF] Kato | Fractional powers of dissipative operators[END_REF][START_REF] Komatsu | Fractional powers of operators[END_REF] and the references therein. In the last years, this theory has gained a higher interest due to the increasing need for modeling in many scientific fields. In fact, many models issued from Biology, Ecology, Finance, etc, lead to nonlinear problems driven by fractional Laplace-type operators.

The basic example of these operators is the fractional Laplacian. In contrast to the Laplacian which is a local operator, the fractional Laplacian is a paradigm of the vast family of nonlocal operators, and this has immediate consequence in the formulation of basic questions such as the Poisson problem. There is an extensive literature devoted to this topic, contributions include [START_REF] Di Nezza | Valdinoci : Hitchhiker's guide to the fractional Sobolev spaces[END_REF][START_REF] Bisci | Variational methods for nonlocal fractional problems[END_REF][START_REF] Vázquez | The mathematical theories of diffusion[END_REF][START_REF] Bucur | Nonlocal diffusion and applications[END_REF][START_REF] Stinga | User's guide to the fractional Laplacian and the method of semigroups[END_REF][START_REF] Caffarelli | An extension problem related to the fractional Laplacian[END_REF][START_REF] Caffarelli | Fractional elliptic equations, Caccioppoli estimates and regularity[END_REF][START_REF] Duo | Accurate numerical methods for two and three dimensional integral fractional Laplacian with applications[END_REF][START_REF] Duo | A comparative study on nonlocal diffusion operators related to the fractional Laplacian[END_REF][START_REF] Garbaczewski | Fractional Laplacians in bounded domains : Killed, reflected, censored and taboo Lévy flights[END_REF][START_REF] Lischke | What is the fractionnal Laplacian ?[END_REF] and the references therein. Needless to say, these references do not exhaust the rich literature on the subject.

From a probabilistic point of view, the fractional Laplacian is related to anomalous diffusion, that employs fractional order derivatives ; it represents the infinitesimal generator of Lévy stable diffusion process, as it is well known in the theory of probability. For more details, see for instance [START_REF] Bogdan | Censored stable processes[END_REF][START_REF] Sato | Lévy Processes and Infinitely Divisible Distributions[END_REF][START_REF] Garbaczewski | Fractional Laplacians in bounded domains : Killed, reflected, censored and taboo Lévy flights[END_REF][START_REF] Song | Potential theory of subordinate killed Brownian motion in a domain[END_REF][START_REF] Chen | Two-sided eigenvalue estimates for subordinate processes in domains[END_REF] and the references therein. On the other hand, it is possible to obtain a fractional heat equation as a limit of a random walk with long jumps, we refer the interested reader to the excellent note [START_REF] Valdinoci | From the long jump random walk to the fractional Laplacian[END_REF].

As is well known, there are several ways of defining this operator in the whole space R N . As far as we know, there are ten definitions in the literature. It turns out that all these definitions are equivalent (see for instance [START_REF] Kwaśnicki | Ten equivalent definitions of the fractional Laplace operator[END_REF]). On the other hand, several different fractional Laplacians can be defined on an open subset Ω = R N . These alternatives correspond to different ways of taking into account the information from the boundary and the exterior of domain.

The first goal of this work is to present an up-to-date survey on the fractional Laplacian in R N , and on the different fractional Laplacians on an open subset Ω = R N . More precisely, we will present four fractional Laplacians. Two of them are well known and have been widely studied, while the other two have only recently been introduced in the literature. The second goal is to highlight the difference between these four operators. To this end, we will first give some simple and significant examples to illustrate this difference; then we will investigate it numerically. For the sake of simplicity, we will only consider homogeneous Dirichlet boundary-value problems, which will allow us to focus on the main features and subtleties of the theory.

Let us emphasize that the originality of this work does not lie so much in the content, for which we have drawn inspiration from existing literature -notably that cited in the bibliography -as in the presentation and choice of the themes.

We do not claim to make significant additions to a theory that has changed little for several years; but we justify our presentation as a new and quick insight for new readers who wish to have a global and quick understanding of the subject before going deeper.

The rest of paper is composed of four sections. Section 2 is devoted to a review of several (and equivalent) definitions of the fractional Laplacian (-∆) s in R N and its main properties. For the sake of brevity and clarity of exposition, we have chosen to treat only the case 0 < s < 1. In section 3, we briefly recall some properties of fractional Sobolev spaces on an open bounded subset Ω of R N . In Section 4, we expose in some detail four different "fractional Laplacians" : regional (in Subsection Needless to say, it is never pleasant for a mathematician not to give the demonstrations. Nevertheless, in order to keep this paper reasonably short, we will mainly state and comment results without reproducing the proofs. Accordingly, we have tried to give precise references that allow the reader to find them. Hence our rather long bibliography : it is certainly not exhaustive, but hopefully rich enough to track most relevant results.

Notations :

Let N be a positive integer,

x = (x 1 , . . . , x N ) ∈ R N , ξ = (ξ 1 , . . . , ξ N ) ∈ R N and α = (α 1 , . . . , α N ) ∈ N N .
Throughout this paper, we denote by :

• Ω ⊂ R N an open, bounded with smooth boundary ;

• D α := ∂ |α| ∂ α1 x1 . . . ∂ α N x N
where

|α| := α 1 + • • • + α N ; • x α := x α1 1 . . . x α N N ; • (x | ξ) := N k=1
x k ξ k the euclidean inner product of R N ;

• x 2 := N k=1 x 2 k ; • ∆u = N k=1 ∂ 2 u ∂x 2 k ; • u L p (R N ) := R N |u(x)| p dx 1 p
where p ∈ [1, +∞) ;

• B(x, r) := {y ∈ R N ; xy < r} where r ∈ (0, +∞) ;

• C 0 (R N ) := {u ∈ C(R N ) ; u(x) → 0 as x → +∞} ; • C ∞ c (R N ) := {u ∈ C ∞ R N ; u with compact support } ; • C ∞ c (Ω) := {u ∈ C ∞ (Ω) ; u with compact support in Ω} ; • S R N := {ϕ ∈ C ∞ R N ; ∀α, β ∈ N N , sup x∈R N x β D α ϕ(x) < +∞} ; • Fu(ξ) := 1 (2π) N 2
R N e -i(ξ|x) u(x)dx the Fourier transform of u ;

• F -1 u(x) := 1 (2π) N 2
R N e i(ξ|x) u(ξ)dξ ;

• Γ(z) = +∞ 0 t z-1 e -t dt, z ∈ (0, +∞) , the Gamma function.

2.

Fractional Laplacian operator in R N .

Let s ∈ (0, 1). In this section, we will introduce the fractional Laplacian (-∆) s in the whole-space R N and its basic properties. To the best of our knowledge, there are ten equivalent definitions of the fractional Laplacian in the whole-space R N , see e.g. [START_REF] Kwaśnicki | Ten equivalent definitions of the fractional Laplace operator[END_REF]. Here, we will give only five definitions, namely the most frequently used.

2.1.

Fractional Sobolev spaces H s (R N ). As will appear later (see Proposition 2.6 below), the fractional Sobolev space H s (R N ) is the right framework to define the fractional Laplacian in R N . For the sake of completness and the ease of reader, we now recall two equivalent definitions of this space.

First, we define the fractional Sobolev space H s (R N ) through the so-called Gagliardo semi-norm :

[u] H s (R N ) := R N R N |u(x) -u(y)| 2 x -y N +2s dxdy 1 2
Then,

H s (R N ) := {u ∈ L 2 (R N ) ; [u] H s (R N ) < +∞}. For u, v ∈ H s (R N ), we set u, v H s (R N ) := R N u(x)v(x)dx + R N R N (u(x) -u(y))(v(x) -v(y))
xy N +2s dxdy.

One can easily check that •, • H s (R N ) is an inner product on H s (R N ) and that the space H s (R N ) endowed with this inner product is a Hilbert space.

Second, the space H s (R N ) can be defined alternatively via the Fourier transform. We denote

H s (R N ) := u ∈ L 2 (R N ) ; R N (1 + ξ 2s )|Fu(ξ)| 2 dξ < +∞ .
The fractional Sobolev space H s (R N ) coincides with H s (R N ) (see for instance [START_REF] Di Nezza | Valdinoci : Hitchhiker's guide to the fractional Sobolev spaces[END_REF]Proposition 3.4] or [START_REF] Bisci | Variational methods for nonlocal fractional problems[END_REF]Corollary 1.15]).

Definitions of fractional Laplacian and main properties.

In this paragraph, we will give some equivalent definitions of fractional Laplacian in R N and its main properties.

First definition.

The first definition of the fractional Laplacian (-∆) s (s ∈ (0, 1)) is expressed via the Fourier transform.

Let u ∈ S(R N ). We have that Fu ∈ S(R N ), but the function ξ → ξ 2s Fu(ξ) / ∈ S(R N ) because ξ 2s creates a singularity at ξ = 0. On the other hand, ξ 2s Fu(ξ) ∈

L 1 (R N ) ∩ L 2 (R N
), so we can use the inverse Fourier transform. Therefore, we have the following definition :

For any u ∈ S(R N ), we define the operator (-∆)

s : S(R N ) → L 2 (R N ) by (-∆) s u(x) = F -1 ( ξ 2s (Fu)(ξ))(x) ∀x ∈ R N , (2.1)
This operator is called the fractional Laplacian of order s.

First properties : Now, we give some elementary properties of the fractional Laplacian (see for instance [START_REF] Stinga | User's guide to the fractional Laplacian and the method of semigroups[END_REF]). Before this, let us recall that S(R N ) ⊂ C 0 (R N ) and the inclusion is strict.

Let u ∈ S(R N ), we have :

• (-∆) 0 u = u ; • (-∆) 1 u = -∆u ; • for any 0 < s 1 , s 2 < 1, (-∆) s1 • (-∆) s2 u = (-∆) s2 • (-∆) s1 u ; • for any multi-index α ∈ N N , D α (-∆) s u = (-∆) s D α u. In particular, (-∆) s u ∈ C ∞ (R N ). 2.2.2. Second definition.
The second definition of the fractional Laplacian is often used in the literature. It is also the most adequate to define the regional fractional Laplacian (see Definition For any u ∈ S(R N ),

(-∆) s u(x) := C(N, s) lim ε→0 + R N \B(x,ε) u(x) -u(y) x -y N +2s dy ∀x ∈ R N , (2.2)
where (ii) Moreover C(N, s) ∼ s(1 -s) as s → 0 + and s → 1 -.

C(N, s) := s4 s π N 2 Γ(s + N 2 ) Γ(1 -s) . ( 2
(2.4)

For more details on the asymptotic behavior of C(N, s), the interested reader is referred to [38, Section 4] (see also [START_REF] Bisci | Variational methods for nonlocal fractional problems[END_REF]Section 1.3.1]).

We recall that by definition :

P.V. R N u(x) -u(y)
xy N +2s dy := lim

ε→0 + R N \B(x,ε) u(x) -u(y)
xy N +2s dy.

In other words, P.V. stands for the "Cauchy principal value". Since then, we can write

(-∆) s u(x) := C(N, s)P.V. R N u(x) -u(y) x -y N +2s dy ∀x ∈ R N . (2.5)
The following theorem (see for instance [START_REF] Stinga | User's guide to the fractional Laplacian and the method of semigroups[END_REF]Theorem 1]) shows that we can do without using the "Cauchy principal value" in (2.5

) if s ∈ (0, 1 2 ). Theorem 2.1. Let u ∈ S(R N ) and x ∈ R N . The integral R N u(x) -u(y) x -y N +2s dy (2.6)
is absolutely convergent if and only if s ∈ (0, 1 2 ).

So we have

(-∆) s u(x) = C(N, s) R N u(x) -u(y)
xy N +2s dy, for s ∈ (0, 1 2 ).

Third definition.

The third definition is given in the form of a well-defined integral on R N as follows :

For any u ∈ S(R N ), we have Another definition of the fractional Laplacian comes from the theory of semigroups in R N × [0, +∞). This definition is classical (see, e.g., [START_REF] Balakrishnan | Fractional powers of closed operators and the semigroups generated by them[END_REF][START_REF] Kato | Fractional powers of dissipative operators[END_REF][START_REF] Komatsu | Fractional powers of operators[END_REF][START_REF] Yosida | Functional Analysis[END_REF]) and it is more useful to study partial differential equations.

(-∆) s u(x) := - 1 2 C(N, s) R N u(x + y) + u(x -y) -2u(x) y N +2s dy, ∀x ∈ R N . (2.
For any u ∈ S(R N ), we have

(-∆) s u(x) = s Γ(1 -s) +∞ 0 (u(x) -w(x, t)) dt t 1+s , (2.8)
where w(x, t) is the solution to the following problem

∂ t w -∆w = 0 on R N × [0, +∞), w(x, 0) = u(x).
All the definitions above are equivalent. For a proof of the equivalence between 2.2 and 2.8, see for instance [81, Lemma 2.1].

2.2.5. Relation between the "classical" Laplacian and the fractional Laplacian.

The following proposition describes the relation between the "classical" Laplacian and the fractional Laplacian (see, e.g., [START_REF] Bucur | Nonlocal diffusion and applications[END_REF]Lemma 3.11] and [START_REF] Di Nezza | Valdinoci : Hitchhiker's guide to the fractional Sobolev spaces[END_REF]Proposition 4.4]).

Proposition 2.1. Let u ∈ S(R N ). Then (i) lim s→0 + (-∆) s u = u, (ii) lim s→1 - (-∆) s u = -∆u.
As we will see later, Proposition 2.1 holds for less regular functions, see Theorem we can extend these definitions to C 0,β (R N ) and even to H s (R N ).

• Extension to C 0,β (R N ) and some regularity results

Let

V be an open subset of R N , k ∈ N and β ∈ (0, 1]. We denote by C 0,β (V ) the space of β-Hölder continuous functions on V , and by C k,β (V ) the space of functions of C k (V ) whose kth partial derivatives are Hölder continuous with exponent β. Let us recall the seminorms :

[u] C 0,β (V ) := sup x,y∈V x =y |u(x) -u(y)| x -y β and [u] C k,β (V ) := |α|=k sup x,y∈V x =y |D α u(x) -D α u(y)| x -y β , k > 0.
Furthermore, let introduce the following space

L 1 s (R N ) = u : R N → R ; R N |u(x)| 1 + x N +2s dx < +∞ .
We have the following results whose proofs can be found in [77, Propositions 2.4, 2.5, 2.6 and 2.7] (see also [START_REF] Silvestre | Regularity of the obstacle problem for a fractional power of the Laplace operator[END_REF] and [80, section 7]). [START_REF] Abatangelo | Large solutions for fractional Laplacian Operators[END_REF], for some ε > 0. Then, (-∆) s u is a continuous function in V and (-∆) s u(x) is given by the definitions above for every x ∈ V .

Proposition 2.2. Let u ∈ L 1 s (R N ). Assume that u ∈ C 0,2s+ε (V ) if s ∈ (0, 1 2 ) or u ∈ C 1,2s+ε-1 (V ) if s ∈ [ 1 2 ,
Proposition 2.3. Let u ∈ C 0,β (R N ) for β > 2s > 0. Then, (-∆) s u ∈ C 0,β-2s (R N )
and there exists C > 0 such that

[(-∆) s u] C 0,β-2s (R N ) ≤ C[u] C 0,β (R N ) ,
where C depends only on β, s and N .

Proposition 2.4. Let u ∈ C 1,β (R N ). (i) If β > 2s, then (-∆) s u ∈ C 1,β-2s (R N ) and there exists C > 0 such that [(-∆) s u] C 1,β-2s (R N ) ≤ C[u] C 1,β (R N ) ,
where C depends only on β, s and N .

(ii) If min(0, 2s -1) < β < 2s, then (-∆) s u ∈ C 0,β-2s+1 (R N ) and there exists C > 0 such that [(-∆) s u] C 0,β-2s+1 (R N ) ≤ C[u] C 1,β (R N ) ,
where C depends only on β, s, and N .

Proposition 2.5. Let u ∈ C k,β (R N ) such that k + β -2s is not an integer. Then,

(-∆) s u ∈ C ,α (R N )
where is the integer part of k + β -2s and α = k + β -2s -.

Before ending this paragraph, it is worth to point out that Proposition 2.1 can be extended for less regular functions. More precisely, we have the following two results (see [START_REF] Stinga | User's guide to the fractional Laplacian and the method of semigroups[END_REF]Theorem 3] and [START_REF] Stinga | User's guide to the fractional Laplacian and the method of semigroups[END_REF]Theorem 4]).

Theorem 2.2. Let x ∈ R N and u ∈ C 2 (B(x, 1)) ∩ L ∞ (R N ). Then lim s→1 -(-∆) s u(x) = -∆u(x). Theorem 2.3. Let x ∈ R N and u ∈ C 0,β (B(x, 1)) ∩ L 0 (R N ) where L 1 0 (R N ) := u : R N → R measurable ; R N |u(x)| 1 + x N dx < +∞ . Then lim s→0 + (-∆) s u(x) = u(x). •• Extension to H s (R N ) Since the Schwartz space S(R N ) is dense in H s (R N ),
(R N ), [u] 2 H s (R N ) = 2C(N, s) -1 (-∆) s 2 u 2 L 2 (R N ) .
(2.9)

As mentioned before, the identity (2.9) shows that the space H s (R N ) is the right framework to define the fractional Laplacian (-∆) s .

Fifth definition.

An important tool in some works on nonlinear problems driven by fractional Laplacians (see, e.g., [START_REF] Silvestre | Regularity of the obstacle problem for a fractional power of the Laplace operator[END_REF][START_REF] Caffarelli | An extension problem related to the fractional Laplacian[END_REF][START_REF] Silvestre | Regularity of the obstacle problem for a fractional power of the Laplace operator[END_REF][START_REF] Caffarelli | Fractional elliptic equations, Caccioppoli estimates and regularity[END_REF]) is the following transformation : any fractional Laplacian can be characterized as an operator that maps a Dirichlet boundary condition to a Neumann-type condition via an extension problem. More precisely, the fractional Laplacian can be realized in a local manner through the boundary

value problem div (y a ∇U ) = 0 in R N +1 + , U (x, 0) = u(x) on ∂R N +1 + , (2.10)
where

R N +1 + := (x, y) ∈ R N × R ; y > 0 is the upper half-space and ∂R N +1 + := R N .
The parameter a belongs to (-1, 1) and is related to the power s of the fractional Laplacian (-∆) s by the formula a = 1 -2s ∈ (-1, 1). We set ∂U ∂ν a := -lim y→0 + y a ∂ y U. The following formula relating the fractional Laplacian to the Dirichlet-to-Neumann operator for (2.10) has been proven in [START_REF] Caffarelli | An extension problem related to the fractional Laplacian[END_REF] (cf. also [START_REF] Brändle | A concave convex elliptic problem involving the fractional Laplacian[END_REF][START_REF] Capella | Regularity of radial extremal solutions for some non-local semilinear equations[END_REF][START_REF] Stinga | Extension problem and Harnack's inequality for some fractional operators[END_REF]) :

(-∆) s u = d s ∂U ∂ν a in R N ,
where d s = 2 2s-1 Γ(s) Γ(1-s) (see e.g. [START_REF] Cabré | Nonlinear equations for fractional Laplacians I : Regularity, maximum principles, and Hamiltonian estimates[END_REF] for more details). -First, we define the fractional Sobolev space H s (Ω) through the Gagliardo semi-norm :

[u] H s (Ω) := Ω Ω |u(x) -u(y)| 2 x -y N +2s dxdy 1 2 . (3.1) Then, H s (Ω) := {u ∈ L 2 (Ω) ; [u] H s (Ω) < +∞}. For u, v ∈ H s (Ω), we set u, v H s (Ω) := Ω u(x)v(x)dx + Ω Ω (u(x) -u(y))(v(x) -v(y))
xy N +2s dxdy.

It is easy to check that •, • H s (Ω) is an inner product and the space H s (Ω) endowed with this inner product is a Hilbert space.

From now on, we denote

u H s (Ω) := Ω |u(x)| 2 dx + Ω Ω |u(x) -u(y)| 2 x -y N +2s dxdy 1 2 . ( 3.2)
-Furthermore, we define H s 0 (Ω) as the closure of

C ∞ c (Ω) in H s (Ω).
-It is worth to notice that the spaces H s (Ω) have no trace where s ∈ (0, 1 2 ]. More precisely, we have the following result : -If s ∈ (0, 1 2 ], then H s 0 (Ω) = H s (Ω).

-If s ∈ ( For this purpose, let us introduce the following space

X s 0 (Ω) := {u ∈ H s (R N ) ; u = 0 a.e. in R N \ Ω} endowed with the norm induced by . H s (R N ) . Let u ∈ X s 0 (Ω), we have u 2 X s 0 (Ω) = u 2 H s (R N ) = u 2 L 2 (R N ) + R N R N |u(x) -u(y)| 2
xy N +2s dxdy.

As u 2 L 2 (R N ) = u 2 L 2 (Ω) , then u 2 X s 0 (Ω) = u 2 L 2 (Ω) + R N R N |u(x) -u(y)| 2
xy N +2s dxdy.

We deduce from (3.3) that the norms • X s 0 (Ω) and • H s (Ω) are not the same.

Moreover, the norm • X s 0 (Ω) takes into account interaction between Ω and R N \ Ω.

So, the space X s 0 (Ω) is the appropriate space to deal with elliptic partial differential equations driven by fractional laplacian operators. Now, for u ∈ X s 0 (Ω), we set

|u | X s 0 (Ω) := R N R N |u(x) -u(y)| 2 x -y N +2s dxdy 1 2
.

We have the following result (see, e.g., [62, Lemma 1.29]) :

Proposition 3.1. The space X s 0 (Ω), •, • X s 0 (Ω) is a Hilbert space with the inner product u, v X s 0 (Ω) := R N R N (u(x) -u(y))(v(x) -v(y)) x -y N +2s dxdy ∀u, v ∈ X s 0 (Ω).
Moreover , the norms | • | X s 0 (Ω) and • X s 0 (Ω) are equivalent.

Remark 3.1. This space is also denoted H s (Ω) or H s 00 (Ω) in the literature. As is well-known (see, e.g., [START_REF] Lions | Non-homogeneous Problems and Applications[END_REF][START_REF] Grisvard | Elliptic Problems in Nonsmooth Domains[END_REF][START_REF] Adams | Sobolev spaces[END_REF]), it holds that

X s 0 (Ω) = H s (Ω) if 0 < s < 1 2 , H s 0 (Ω) if 1 2 < s < 1,
while in the limiting case s =1 2 , it holds that :

X 1 2 0 (Ω) := u ∈ H 1 2 (Ω) ; Ω u 2 (x) ρ(x) dx < +∞ , with ρ(x) = distance(x, ∂Ω).
In all cases, the canonical norm induced by the alternative definition is equivalent to the previous ones. In this section, we will present four fractional Laplacians and compare their definitions, their probabilistic interpretations and their analytical properties.

For pedagogical purposes, this section is composed of six subsections. In the first one, we introduce the regional fractional Laplacian (FL Reg for short) and give some examples. Subsection 2 is devoted to the spectral fractional Laplacian (FL Spec ). In subsection 3, we highlight the difference between these two operators by comparing, in particular, their respective spectra. In Subsection 4, we introduce the so-called restricted fractional Laplacian (FL Rest ), and we compare it with FL Reg . Then, we introduce the so-called peridynamic fractional Laplacian (FL Pery ) in Subsection 5.

In the last subsection, we give three examples in dimension 1 in order to illustrate the difference between these four operators.

In order to go deeper into the study of fractional Laplacian operators, we refer the reader, for instance, to [START_REF] Leonori | Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations[END_REF][START_REF] Biccari | Local elliptic regularity for the Dirichlet fractional Laplacian[END_REF][START_REF] Biccari | Local regularity for fractional heat equations[END_REF]. Those papers give basic estimates and some regularity results for solutions to problems governed by (-∆) s .

On the other hand, the paper [START_REF] Claus | Realization of the fractional Laplacian with nonlocal exterior conditions via forms method[END_REF] gives, through the theory of semigroups, a characterization of the realization in L2 (Ω) of the fractional Laplacian with different exterior conditions.

4.1. Regional Fractional Laplacian (FL Reg ).

In this subsection, we give the definition of the so-called regional fractional Laplacian which we denote by (-∆) s Reg . We also give a few examples for illustration purposes.

• Definition This operator acts on functions u defined in Ω and extended by zero to R N \ Ω. So we can use Definition 2.2.

Let u ∈ X s 0 (Ω). The regional fractional Laplacian 2 is defined by : (-∆) s Reg u(x) := C(N, s)P.V.

R N u(x) -u(y)
xy N +2s dy. xy N +2s dy .

As u = 0 ∈ R N \ Ω, we obtain (-∆) s Reg u(x) := C(N, s) P.V. Ω u(x) -u(y) x -y N +2s dy + R N \Ω u(x) x -y N +2s dy .
(4.2) Formula (4.2) shows that (-∆) s Reg depends on Ω and R N \ Ω.

Probabilistic interpretation : The regional fractional Laplacian is the infinitesimal generator of jump-type processes killed upon leaving the domain Ω. In this respect, the operator generates a killed subordinate Brownian motion (see for instance [START_REF] Bogdan | Censored stable processes[END_REF][START_REF] Chen | Two-sided eigenvalue estimates for subordinate processes in domains[END_REF][START_REF] Garbaczewski | Fractional Laplacians and Levy flights in bounded domains[END_REF][START_REF] Garbaczewski | Fractional Laplacians in bounded domains : Killed, reflected, censored and taboo Lévy flights[END_REF][START_REF] Song | On the relationship between subordinate killed and killed subordinate processes[END_REF] and references therein). Indeed, this process describes trajectories of particles which are allowed to jump outside Ω and then one subordinates these paths in Ω, i.e., one leaves marks only on the parts of paths inside the domain. The homogeneous Dirichlet exterior condition in R N \ Ω corresponds to this operation.

• Examples For the reader's convenience, let us give some examples, (see [START_REF] Dyda | Fractional calculus for power functions and eigenvalues of the fractional Laplacian[END_REF]).

Example 1. One-dimensional case : Ω = (-1, 1).

u(x) (-∆) s Reg u(x) (1 -x 2 ) s-1 + 0 (1 -x 2 ) s + Γ(2s + 1) (1 -x 2 ) s+1 + (s + 1)Γ(2s + 1)(1 -(2s + 1)x 2 ) (1 -x 2 ) s+2 + (s+1)(s+2) 2 Γ(2s + 1)(1 -(4s + 2)x 2 + ( 2s 3 + 1)(2s + 1)x 4 ) x(1 -x 2 ) s-1 + 0 x(1 -x 2 ) s + Γ(2s + 2)x x(1 -x 2 ) s+1 + Γ(2s+3) 6 (3 -(2s + 3)x 2 )x x(1 -x 2 ) s+2 + s+2 60 Γ(2s + 3)(15 -(20s + 30)x 2 + (2s + 3)(2s + 5)x 4 )x
Table 1. The regional fractional Laplacian of some functions that vanish in R \ (-1, 1).

Example 2. Multidimensional case : Let N ≥ 2, x = (x 1 , x 2 , ..., x N ) ∈ R N and Ω = B(0, 1) be the ball of center the origin and radius 1.

u(x) (-∆) s Reg u(x) (1 -x 2 ) s + 4 s Γ(s + 1)Γ( 2s+N 2 )Γ( N 2 ) -1 (1 -x 2 ) s+1 + 4 s Γ(s + 2)Γ( 2s+N 2 )Γ( N 2 ) -1 (1 -(1 + 2s N ) x 2 ) (1 -x 2 ) s + x N 4 s Γ(s + 1)Γ( 2s+N 2 + 1)Γ( N 2 + 1) -1 x N (1 -x 2 ) s+1 + x N 4 s Γ(s + 2)Γ( 2s+N 2 + 1)Γ( N 2 + 1) -1 (1 -(1 + 2s N +2 ) x 2 )
Table 2. The regional fractional Laplacian of some functions that vanish in R N \ B(0, 1).

• Spectrum of FL Reg

It is shown (see, e.g., [START_REF] Servadei | On the spectrum of two different fractional operators[END_REF], [START_REF] Bisci | Variational methods for nonlocal fractional problems[END_REF]Chapter 3]) that, thus defined, (-∆) s Reg : X s 0 (Ω) → L 2 (Ω) is self-adjoint operator with a discrete spectrum and positive eigenvalues,

more precisely 0 < µ 1,s ≤ µ 2,s ≤ • • • ≤ µ k,s ≤ µ k+1,s ≤ • • •
From now on, we denote by {e R k,s } k≥1 the corresonding eigenfunctions normalized

in L 2 (Ω) (i.e. ||e R k,s || L 2 (Ω) = 1).
It is worth to point out that the {e R k,s } k≥1 are (only) Hölder continuous up the boundary, namely e R k,s ∈ C 0,s (Ω) (see, e.g., [START_REF] Chen | Two-sided eigenvalue estimates for subordinate processes in domains[END_REF], [START_REF] Ros-Oton | The Dirichlet problem for the fractional Laplacian: regularity up to the boundary[END_REF]).

Spectral Fractional Laplacian (FL Spec ).

In this subsection, we will give two (equivalent) definitions of the spectral fractional Laplacian and we denote it simply by (-∆) s Spec . The first one is given via the spectrum of the "classical" Laplacian on the same domain, see for instance [START_REF] Servadei | On the spectrum of two different fractional operators[END_REF], [62,

Chapter 5], [START_REF] Chen | Two-sided eigenvalue estimates for subordinate processes in domains[END_REF], [START_REF] Abatangelo | Nonhomogeneous boundary conditions for the spectral fractional Laplacian[END_REF], [START_REF] Caffarelli | Fractional elliptic equations, Caccioppoli estimates and regularity[END_REF], [START_REF] Bonforte | Existence, uniqueness and asymptotic behavior for fractional porous medium equations on bounded domains[END_REF] and the references therein. The second one is given by an integral formula via the semigroups, see e.g. [START_REF] Stinga | Fractional powers of second order partial differential operators: extension problem and regularity theory[END_REF], [START_REF] Vázquez | The mathematical theories of diffusion[END_REF] and the references therein.

• First definition : Let {λ k } k∈N * with 0 < λ 1 < λ 2 ≤ ... be the divergent sequence of eigenvalues of the "classical" Laplacian -∆ in Ω with homogeneous Dirichlet boundary data on ∂Ω, and {e k } k∈N * the corresponding eigenfunctions, that is,

   -∆e k = λ k e k in Ω, e k = 0 on ∂Ω,
normalized in such a way that e k L 2 (Ω) = 1 (see, e.g., [START_REF] Henrot | Extremum problems for eigenvalues of Elliptic operators[END_REF]). Now let us introduce the following space defined by

H s (Ω) := {u = +∞ k=1 u k e k ∈ L 2 (Ω) ; +∞ k=1 λ s k u 2 k < +∞}. (4.3)
We easily check that ( H s (Ω), •, • H s (Ω) ) is a Hilbert space, where

u, v H s (Ω) := +∞ k=1 λ s k u k v k , for u = +∞ k=1 u k e k and v = +∞ k=1 v k e k belong H s (Ω) .
It turns out that H s (Ω) = X s 0 (Ω), see for instance [16, section 3.1.3] and the references therein. Now we are ready to state the definition. More precisely :

for u = +∞ k=1
u k e k ∈ X s 0 (Ω), the spectral fractional Laplacian, denoted by (-∆) s Spec , is defined by

(-∆) s Spec u(x) := +∞ k=1 u k λ s k e k (x), (4.4)
• Spectrum of FL Spec Thanks to (4.4), the eigenvalues and the eigenfunctions of FL Spec are, respectively, λ s k and e k for all k ∈ N * .

In order to avoid any confusion, from here on we denote by {e S k,s , λ k,s } k∈N * the eigenfunctions and the eigenvalues of (-∆) s Spec on Ω in the rest of this paper.

Example 3.

-Let Ω = (0, 1) and u(x) = sin(πx). Then, (-∆) s Spec u(x) = π 2s sin(πx).

-Let Ω = (-1, 1) and u(x) = sin π(1+x)

2

. Then,

(-∆) s Spec u(x) = π 2 2s sin π(1 + x) 2 .

• Second definition

As in the case of R N (see Definition 2.8), the spectral fractional Laplacian can also be defined through the heat semigroup (see, e.g., [START_REF] Stinga | Fractional powers of second order partial differential operators: extension problem and regularity theory[END_REF] and the references therein).

This technique has been efficiently used in recent research papers (see for instance [START_REF] Cusimano | Discretizations of the spectral fractional Laplacian on general domains with Dirichlet, Neumann, and Robin boundary conditions[END_REF][START_REF] Cusimano | Numerical approximations for fractional elliptic equations via the method of semigroups[END_REF][START_REF] Nochetto | A PDE approach to space-time fractional parabolic problems[END_REF]). More precisely, we have :

(-∆) s Spec u(x) = s Γ(1 -s) +∞ 0 (u(x) -e t∆ u(x)) dt t 1+s , x ∈ Ω, (4.5)
where e t∆ u denotes the solution to the heat equation in Ω × [0, +∞) with initial datum u.

• Probabilistic interpretation : Roughly speaking, the spectral fractional Laplacian generates a subordinate killed Brownian motion, i.e. the process that first kills Brownian motion in Ω and then subordinates it via a s-stable subordinator. The killed process is generated by the classical Laplacian -∆ with homogeneous Dirichlet boundary conditions, and the subordination provides its fractional power (-∆) s Spec . See, e.g., [START_REF] Bogdan | Censored stable processes[END_REF][START_REF] Chen | Two-sided eigenvalue estimates for subordinate processes in domains[END_REF][START_REF] Garbaczewski | Fractional Laplacians and Levy flights in bounded domains[END_REF][START_REF] Garbaczewski | Fractional Laplacians in bounded domains : Killed, reflected, censored and taboo Lévy flights[END_REF][START_REF] Guan | Boundary problems for fractional Laplacians[END_REF][START_REF] Song | On the relationship between subordinate killed and killed subordinate processes[END_REF] and the references therein.

4.3.

Comparison of the operators FL Spec and FL Reg .

The main goal of this paragraph is to show that the operators FL Spec and FL Reg are different.

First : As we have just seen in the previous two paragraphs, the definition itself of (-∆) s Spec only depends on the domain Ω considered, while that of (-∆) s Reg depends on the domain Ω and on R N \ Ω (see (4.2)).

Second : We state below three results showing that the spectra of (-∆) s Spec and (-∆) s Reg are indeed different. For more details, the interested reader is referred to [START_REF] Servadei | On the spectrum of two different fractional operators[END_REF] or [62, Chapter 5] and the references therein.

Ω (at least e S k,s ∈ C ∞ (Ω) ∩ C 2 (Ω)).
Furthermore, the operators (-∆) s Reg and (-∆) s Spec have different eigenvalues.

More precisely, we have these two following results :

Theorem 4.2. [73, Theorem 1] We recall that µ 1,s (resp. λ 1,s ) is the first eigenvalue of FL Reg (resp. FL Spec ). Then,

µ 1,s < λ 1,s .
cλ k,s ≤ µ k,s ≤ λ k,s , ∀k ∈ N * , k ≥ 2.
In particular, C = 1 2 when Ω is convex.

We refer also to [START_REF] Musina | On fractional Laplacians[END_REF] and [START_REF] Musina | On fractional Laplacians -2[END_REF] .

4.4.

Restricted Fractional Laplacian (FL Rest ).

In this subsection, we will introduce another fractional Laplacian operator called restricted fractional Laplacian, 3 In what follows, we will highlight these differences :

First: Unlike the regional fractional Laplacian, the operator FL Rest is generated by symmetric stable processes describing motions of random particles in Ω which are allowed to jump inside Ω, but are either reflected in Ω or killed when they reach the boundary ∂Ω (see for instance [START_REF] Bogdan | Censored stable processes[END_REF][START_REF] Chen | Two-sided heat kernel estimates for censored stable-like processes[END_REF][START_REF] Guan | Boundary problems for fractional Laplacians[END_REF][START_REF] Song | On the relationship between subordinate killed and killed subordinate processes[END_REF]).

Second: Now, we will illustrate their difference analytically.

Let u be a smooth function such that u(x) = 0 for x ∈ R \ Ω. Let x ∈ Ω. Thanks to formula (4.2), we have (-∆) s Reg u(x) = C(N, s)P.V.

R N u(x) -u(y)
xy N +2s dy = C(N, s) P.V.

Ω u(x) -u(y) x -y N +2s dy + R N \Ω u(x) x -y N +2s dy So D 1 (x) = (-∆) s Reg -(-∆) s Rest u(x) = u(x) R N \Ω 1 x -y N +2s dy (4.7)
Now, for the reader's comfort, we explicitly calculate the difference (4.7) in the case Ω = (-L, L) with L > 0. Then we have

D 1 u(x) = C(1, s) -L -∞ 1 |x -y| 1+2s dy + +∞ L 1 |x -y| 1+2s dy u(x).
Thus, for any x ∈ Ω,

D 1 u(x) = C(1, s) 2s 1 (x + L) 2s + 1 (L -x) 2s u(x). (4.8)
As lim s→1 -C(1, s) = 0 (see (2.4)), we deduce from (4.8) that lim s→1 -D 1 u(x) = 0. Consequently, FL Rest can be used to approximate FL Reg as s → 1 -.

Applying again (2.4), then lim

s→0 + C(1, s) 2s = 1 2 . So lim s→0 + D 1 u(x) = u(x)
. Thus, when s → 0 + , FL Reg can be written as the sum of FL Rest and the identity operator.

To end this sub-paragraph, it is worth pointing out that unlike the regional and spectral fractional Laplacians, the operator FL Rest has not been studied much. For example, nothing is known about its eigenvalues and eigenfunctions.

4.5. Peridynamic Fractional Laplacian (FL Pery ).

The peridynamic model was originally proposed as a reformulation of the classical solid mechanics. We refer the interested reader to the pioneering article [START_REF] Silling | Reformulation of elasticity theory for discontinuities and long-range forces[END_REF], as waell as to [START_REF] Délia | The fractional Laplacian operator on bounded domains as a special case of the nonlocal diffusion operator[END_REF][START_REF] Guan | Analysis and approximation of a nonlocal obstacle problem[END_REF][START_REF] Duo | A comparative study on nonlocal diffusion operators related to the fractional Laplacian[END_REF][START_REF] Alali | Fourier multipliers for nonlocal Laplace operators[END_REF][START_REF] Lischke | What is the fractionnal Laplacian ?[END_REF] and references therein.

Probabilistic interpretation : Let δ ∈ (0, +∞). The peridynamic fractional Laplacian (FL Pery in short) of horizon δ can be seen as the infinitesimal generator of a symmetric 2s-stable Lévy process restricted to B(x, δ)4 for x ∈ Ω. For more details, see, e.g., [START_REF] Lischke | What is the fractionnal Laplacian ?[END_REF], [START_REF] Duo | A comparative study on nonlocal diffusion operators related to the fractional Laplacian[END_REF] and the references therein. The real number δ is called the horizon parameter, and it is often used to measure the range of nonlocal interactions in many nonlocal models. Thence, this operator is sometimes called the "horizonbased nonlocal" operator.

Notation :

In what follows, we will use the following notation (-∆) s,δ P ery or (-∆) s P ery if there is no risk of confusion.

Definition :

Before stating the definition, we need to define the so-called interaction domain of horizon δ, defined by Ω I,δ := y ∈ R N \ Ω ; xy < δ, for x ∈ Ω .

In other words, the domain Ω I,δ consists of those points outside of Ω that interact with points in Ω. In fact, (-∆) s,δ P ery acts on functions u defined in Ω extended by 0

to Ω I,δ . More precisely, (-∆) s,δ P ery is defined by

(-∆) s P ery u(x) = C(N, s)P.V. B(x,δ) u(x) -u(y)
xy N +2s dy. (4.9)

Comparison between FL Reg and FL Pery 1) In the limiting case of δ → ∞, FL Pery coincides with FL Reg , and thus the latter is often used to approximate FL Pery if the parameter δ is sufficiently large (see for example [START_REF] Délia | The fractional Laplacian operator on bounded domains as a special case of the nonlocal diffusion operator[END_REF][START_REF] Guan | Analysis and approximation of a nonlocal obstacle problem[END_REF]).

2) In order to illustrate the difference between these two operators, we will proceed as in the previous paragraph. For this, we choose a smooth function u such that u(x) = 0 for x ∈ R \ Ω with Ω = (-L, L). Here, we assume that the horizon size δ in (4.9) is large enough such that δ > 2L. As is done in the previous paragraph, we compute their difference :

D 2 u(x) = (-∆) s Reg -(-∆) s P ery u(x) = C(1, s)   P.V. R u(x) -u(y) |x -y| 1+2s dy -P.V x+δ x-δ u(x) -u(y) |x -y| 1+2s dy   = C(1, s) x-δ -∞ 1 |x -y| 1+2s dy + +∞ x+δ 1 |x -y| 1+2s dy u(x) So, for all x ∈ Ω D 2 u(x) = (-∆) s Reg -(-∆) s P ery u(x) = C(1, s) sδ 2s u(x), (4.10)
We conclude that the difference of these two operators is of order O( 1 δ 2s ) when u(x) is uniformly bounded on Ω, hence their difference vanishes as δ → ∞. On the other hand, the convergence of FL Pery to FL Reg as δ → ∞ depends on the power s, and it may degenerate rapidly for small s. Additionally, in the limiting case of s → 1 -, the difference D 2 u(x) → 0, because the coefficient C(1, s) → 0 (see (2.4)).

Examples.

After this brief review of pairwise differences between them, let us compare all four operators by applying them to the same functions on Ω = (-1, 1).

Example 1 : Consider the function (see [START_REF] Duo | A comparative study on nonlocal diffusion operators related to the fractional Laplacian[END_REF]) defined by -for small s, the values of (-∆) s Rest are at variance with those of (-∆) s Reg .

u(x) =      sin π(1+x) 2 if x ∈ (-
However, (-∆) s P ery can still provide a good approximation to (-∆) s Reg by enlarging the horizon size δ. where q ∈ N. In this case, (-∆) s Reg u can be computed explicitly

(-∆) s Reg u(x) = 2 2s Γ( 1+2s
2 )Γ(s + q + 1) √ πΓ(q + 1) 2 F 1 1 + 2s 2 , -q ; 1 2 ; x 2 for x ∈ Ω, where 2 F 1 denotes the Gauss hypergeometric function. Moreover, we can obtain the exact values of (-∆) s Rest u and (-∆) s P ery u by using their relation to (-∆) s Reg in (4.8) and (4.10), respectively. For (-∆) s Spec , (-∆) s Spec u is numerically computed by the finite difference method proposed in [START_REF] Duo | A fast algorithm for solving the space-time fractional diffusion equation[END_REF].

In Figure 2, the four functions (-∆) s Spec u, (-∆) s Reg u, (-∆) s P ery u and (-∆) s Rest u are compared. The function u is defined by (4.12) with q = 2.

Figure 2 shows that the functions (-∆) s Spec u, (-∆) s Rest u, (-∆) s P ery u and (-∆) s Reg u exist on the closed domain Ω for any s ∈ (0, 1), but their values are very different, especially for small s. The values of (-∆) s Spec u are always zero at boundary points.

Also, both (-∆) s Rest and (-∆) s P ery with relatively small δ can provide a good approximation to (-∆) s Reg , if s is large. When s is small, however, (-∆) s P ery can be One can check that:

• if L = (-∆) s Reg , u(x) = 2 -2s √ π Γ( 1+2s
2 )Γ(1 + s)

(1 -x 2 ) s for x ∈ (-1, 1) ; On the other hand, the solutions of the nonlocal problem (4.13) with L = (-∆) s Rest or (-∆) s P ery are not analytically known and will be numerically computed.

• if L = (-∆) s Spec , u (
Figure 3 shows the solutions to (4.13) with all four operators. In the case of FL Rest , since the solution does not exist for s ≤ 1 2 , we only present it for s > 1 2 . Generally speaking, these solutions are significantly different. However, as s → 1 -they all converge to the function u(x) = 1 2 (1 -x 2 ), the solution to the classical Poisson equation :

-u (x) = 1, if x ∈ (-1, 1) ; u(-1) = u(1) = 0.

4. 1 )

 1 , spectral (in Subsection 4.2), restricted (in Subsection 4.4) and peridynamic (in Subsection 4.5). Moreover, Subsections 4.3 and 4.6 are devoted to highlighting the differences between these four operators by suitable one-dimensional examples.

4. 1 )

 1 and the restricted fractional Laplacian (see Definition 4.6) in the case of a bounded subset of R N .

3 . 3 . 1 .

 331 Fractional Sobolev spaces on an open bounded subset of R N . In what follows, Ω is an open bounded subset of R N with smooth boundary, and s ∈ (0, 1). As in the case of R N , we start this paragraph by recalling one of the possible definitions of the classical fractional Sobolev spaces H s (Ω) and H s 0 (Ω). Then we define new spaces X s 0 (Ω), which are the appropriate functional framework to study Dirichlet boundary-value problems governed by fractional Laplacians. Classical fractional Sobolev spaces H s (Ω) and H s 0 (Ω).

Theorem 3 . 1 .

 31 [START_REF] Lions | Non-homogeneous Problems and Applications[END_REF] Theorem 11.1]

4 .

 4 Fractional Laplacians on an open bounded domain. Througout this paragraph, Ω is a bounded open subset of R N with smooth boundary. As we have seen in Section 2, the different definitions of the fractional Laplacian (-∆) s are equivalent when the operator acts in R N . Contrary to the case of whole space, several different fractional Laplacians 1 can be defined on a open bounded subset of R N . These alternatives correspond to different ways in which the information coming from the boundary and the exterior of domain is to be taken into account.

(4. 1 ) 4 . 1 .

 141 Remark From (4.1), we deduce(-∆) s Reg u(x) := C(N, s) P.V. Ω u(x) -u(y) xy N +2s dy + R N \Ω u(x) -u(y)

Definitions 4 .

 4 4 and 4.5 are equivalent (see, e.g., [

Theorem 4 . 1 .

 41 [73, Theorem 1.1 and Theorem 1.2] The eigenfunctions e R k,s are only Hölder continuous on Ω. However, the eigenfunctions e S k,s are smooth on

Theorem 4 . 3 .

 43 [START_REF] Chen | Two-sided eigenvalue estimates for subordinate processes in domains[END_REF] Theorem 4.5] There exists a constant C = C(Ω) > 0 such that:

Figure 1 .

 1 Figure 1. Comparison of the functions (-∆) s u where u is defined in (4.11) and (-∆) s represents (-∆) s Spec (--), (-∆) s Reg (----), (-∆) s Rest (-• -•) or (-∆) s P ery (••••••) with δ = 4. The result for -∆ ( * * * * ) is included in the plot of s = 0.975.

Figure 2 .Example 3 :

 23 Figure 2. Comparison of the functions (-∆) s u where u is defined in (4.12) and (-∆) s represents (-∆) s Spec (--), (-∆) s Reg (----), (-∆) s Rest (-• -•) or (-∆) s P ery (••••••) with δ = 4. The result for -∆ ( * * * * ) is included in the plot of s = 0.975.

Figure 3 .

 3 Figure 3. Comparison of the solution to (4.13) with (-∆) s Spec (--), (-∆) s Reg (----), (-∆) s Rest (-• -•) or (-∆) s P ery (••••••) with δ = 4. The result for -∆ ( * * * * ) is included in the plot of s = 0.975.

  7) It is worth to point out that the Definitions 2.1, 2.2 and 2.7 are equivalent. For a proof, see for instance [38, section 3] or [62, subsection 1.3].

2.2.4. Fourth definition.

  1 2 , 1), the inclusion H s 0 (Ω) ⊂ H s (Ω) is strict. The previous result holds for any p ∈ (1, +∞), see [52, Theorem 1.4.2.4]. 3.2. Other type of fractional Sobolev spaces X s 0 (Ω). In order to study nonlocal problems governed by fractional Laplacians, we need appropriate fractional Sobolev spaces. These new spaces are inspired by (but not equivalent to) the fractional Sobolev spaces H s (Ω), in order to correctly encode the data of the homogeneous Dirichlet condition in the weak formulation.

  It goes without saying that the operators FL Rest and FL Reg are different from each other, even though their names and notations may be interchanged in some works.

	u ∈ X s 0 (Ω) by the formula	which we denote (-∆) s Rest . It is defined for any
	(-∆) s Rest u(x) = C(N, s)P.V.	Ω	u(x) -u(y) x -y N +2s dy, x ∈ Ω.	(4.6)

Unfortunately, the mathematical community has not yet reached a unified terminology and standard notations for the fractional Laplacians on a bounded domain. The reader is strongly advised to check the definitions carefully in order to avoid any confusion.

This operator is also called the restricted fractional Laplacian (see e.g.[START_REF] Vázquez | The mathematical theories of diffusion[END_REF][START_REF] Bonforte | Existence, uniqueness and asymptotic behavior for fractional porous medium equations on bounded domains[END_REF][START_REF] Bonforte | Sharp global estimates for local and nonlocal porous medium-type equations in bounded domains[END_REF]), or hypersingular fractional Laplacian (see for instance[START_REF] Duo | Accurate numerical methods for two and three dimensional integral fractional Laplacian with applications[END_REF][START_REF] Garbaczewski | Fractional Laplacians and Levy flights in bounded domains[END_REF]), or standard fractional Laplacian (see for example[START_REF] Cao | Simultaneously recovering potentials and embedded obstacles for anisotropic fractional Schrödinger operators[END_REF]), or Riesz fractional Laplacian see, e.g.,[START_REF] Lischke | What is the fractionnal Laplacian ?[END_REF], or simply fractional Laplacian, ...

As already mentioned, there is no consensus in terminology. Indeed, the operator FL Rest is sometimes called regional (see,e.g,[41, 61,

[START_REF] Fall | Regional Fractional Laplacians: boundary regularity[END_REF] or censored. We have opted for 'restricted' because the values of (-∆) s Rest u only depend on the values of u in Ω, not in the complementary.

Let us recall that B(x, δ) denotes the ball centered at point x with radius δ