ON THE SPECTRUM OF TWO DIFFERENT
FRACTIONAL OPERATORS
RAFFAELLA SERVADEI AND ENRICO VALDINOCI
Abstract. In this paper we deal with two nonlocal operators, that are both well known
and widely studied in the literature in connection with elliptic problems of fractional type.
Precisely, for a fixed s ∈ (0, 1) we consider the integral definition of the fractional Laplacian
given by
(−∆)
s
u(x) :=
c(n, s)
2
Z
R
n
2u(x) − u(x + y) − u(x − y)
|y|
n+2s
dy , x ∈ R
n
,
where c(n, s) is a positive normalizing constant, and another fractional operator obtained
via a spectral definition, that is
A
s
u =
X
i∈N
a
i
λ
s
i
e
i
,
where e
i
, λ
i
are the eigenfunctions and the eigenvalues of the Laplace operator −∆ in Ω
with homogeneous Dirichlet boundary data, while a
i
represents the projection of u on the
direction e
i
.
Aim of this paper is to compare these two operators, with particular reference to their
spectrum, in order to emphasize their differences.
Contents
1. Introduction 1
2. A comparison between the eigenfunctions of A
s
and (−∆)
s
3
2.1. Poisson kernel of fractional type 4
2.2. Optimal regularity for the eigenfunctions of (−∆)
s
4
3. The spectrum of A
s
and (−∆)
s
9
4. One-dimensional analysis 9
5. A relation between the first eigenvalue of A
s
and that of (−∆)
s
14
References 17
1. Introduction
Recently in the literature a great attention has been devoted to the study of nonlocal
problems driven by fractional Laplace type operators, not only for a pure academic interest,
but also for the various applications in different fields. Indeed, many different problems
driven by the fractional Laplacian were considered in order to get existence, non-existence
and regularity results and, also, to obtain qualitative properties of the solutions.
In particular, two notions of fractional operators were considered in the literature, namely
the integral one (which reduces to the classical fractional Laplacian, see, for instance, [7, 8,
9, 10, 14, 15, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33] and references therein) and the
Key words and phrases. Fractional Laplace operator, Laplace operator, eigenvalues and eigenfunctions.
2010 AMS Subject Classification: 35R09, 45K05, 35R11, 26A33.
The first author was supported by the MIUR National Research Project Variational and Topological
Methods in the Study of Nonlinear Phenomena, while the second one by the MIUR National Research
Project Nonlinear Elliptic Problems in the Study of Vortices and Related Topics and the FIRB project A&B
(Analysis and Beyond). Both the authors were supported by the ERC grant ε (Elliptic Pde’s and Symmetry
of Interfaces and Layers for Odd Nonlinearities).
1
2 R. SERVADEI AND E. VALDINOCI
spectral one (that is sometimes called the regional, or local, fractional Laplacian, see, e.g.
[2, 4, 5, 6, 35] and references therein).
For any fixed s ∈ (0, 1) the fractional Laplace operator (−∆)
s
at the point x is defined
by
(1.1) (−∆)
s
u(x) :=
c(n, s)
2
Z
R
n
2u(x) − u(x + y) − u(x − y)
|y|
n+2s
dy ,
where c(n, s) is a positive normalizing constant
1
depending only on n and s .
A different operator, which is sometimes denoted by A
s
, is defined as the power of
the Laplace operator −∆ , obtained by using the spectral decomposition of the Laplacian.
Namely, let Ω be a smooth bounded domain of R
n
, and let λ
k
and e
k
, k ∈ N, be the
eigenvalues and the corresponding eigenfunctions of the Laplacian operator −∆ in Ω with
zero Dirichlet boundary data on ∂Ω, that is
−∆e
k
= λ
k
e
k
in Ω
e
k
= 0 on ∂Ω ,
normalized in such a way that ke
k
k
L
2
(Ω)
= 1 . For any s ∈ (0, 1) and any u ∈ H
1
0
(Ω) with
u(x) =
X
i∈N
a
i
e
i
(x) , x ∈ Ω ,
one considers the operator
(1.2) A
s
u =
X
i∈N
a
i
λ
s
i
e
i
.
Aim of this paper is to compare the two previous definitions of fractional Laplace opera-
tors. First of all, we would like to note that these two fractional operators (i.e. the ‘integral’
one and the ‘spectral’ one) are different (in spite of some confusion that it is possible to
find in some of the existent literature in which the two operators are somehow freely in-
terchanged). Indeed, the spectral operator A
s
depends on the domain Ω considered (since
its eigenfunctions and eigenvalues depend on Ω), while the integral one (−∆)
s
evaluated at
some point is independent on the domain in which the equation is set.
2
Of course, by definition of A
s
, it is easily seen that the eigenvalues and the eigenfunctions
of A
s
are respectively λ
s
k
and e
k
, k ∈ N , that is the s-power of the eigenvalues of the
Laplacian and the very same eigenfunctions of the Laplacian, respectively.
On the other hand, the spectrum of (−∆)
s
may be less explicit to describe. We refer to
[28, Proposition 9 and Appendix A], [23, 24], [25, Proposition 5] and [30, Proposition 4] for
the variational characterization of the eigenvalues and for some basic properties.
A natural question is whether or not there is a relation between the spectrum of A
s
and
(−∆)
s
and, of course, between the respective eigenfunctions. In the present paper, by using
the classical regularity theory for the eigenfunctions of the Laplace operator −∆ and some
recent regularity results for the fractional Laplace equation (see [22, 23, 24, 32]), we will
show that the eigenfunctions of A
s
and (−∆)
s
are different (for more details see Section 2).
In particular, we will show that the eigenfunctions of (−∆)
s
are, in general, no better than
H¨older continuous up to the boundary, differently from the eigenfunctions of A
s
(i.e. of the
classical Laplacian) that are smooth up to the boundary (if so is the domain).
1
Different definitions of the fractional Laplacian consider different normalizing constants. The constant
c(n, s) chosen here is the one coming from the equivalence of the integral definition of (−∆)
s
and the one
by Fourier transform (see, e.g., [7] and [10, (3.1)–(3.3) and (3.8)]) and it has the additional properties that
lim
s→1
−
(−∆)
s
u = −∆u and lim
s→0
+
(−∆)
s
u = u (see [10, Proposition 4.4]).
2
Also, the natural functional domains for the operators (−∆)
s
and A
s
are different, but this is a minor
distinction, since one could consider both the operators as acting on a very restricted class of functions for
which they both make sense - e.g., C
∞
0
(Ω).
ON THE SPECTRUM OF TWO DIFFERENT FRACTIONAL OPERATORS 3
Furthermore, with respect to the eigenvalues of A
s
and (−∆)
s
, we will prove that the
first eigenvalue of (−∆)
s
is strictly less than the first one of A
s
. To this purpose we will
use some extension results for the fractional operators A
s
and (−∆)
s
(see [7, 34]).
Summarizing, the results given in this paper are the following:
Theorem 1. The operators (−∆)
s
and A
s
are not the same, since they have different
eigenvalues and eigenfunctions. In particular:
• the first eigenvalues of (−∆)
s
is strictly less than the one of A
s
;
• the eigenfunctions of (−∆)
s
are only H¨older continuous up to the boundary, dif-
ferently from the ones of A
s
that are as smooth up the boundary as the boundary
allows.
For further comments on similarities and differences between the operators A
s
and (−∆)
s
for s = 1/2 see [13, Remark 0.4].
The paper is organized as follows. Section 2 is devoted to a comparison between the
eigenfunctions of A
s
and (−∆)
s
. In Section 3 we deal with the spectrum of the two fractional
operators we are considering. Section 4 is devoted to the extension of the operator A
s
, while
in Section 5 we discuss the relation between the first eigenvalues of A
s
and (−∆)
s
.
2. A comparison between the eigenfunctions of A
s
and (−∆)
s
This section is devoted to some remarks about the eigenfunctions of the operators A
s
and
(−∆)
s
. Precisely, we will consider the following eigenvalue problems in a smooth bounded
domain Ω ⊂ R
n
, with Dirichlet homogeneous boundary data, driven, respectively, by A
s
and (−∆)
s
,
(2.1)
A
s
u = λu in Ω
u = 0 on ∂Ω
and
(2.2)
(−∆)
s
u = λu in Ω
u = 0 in R
n
\ Ω .
Note that in (2.2) the boundary condition is given in R
n
\ Ω and not simply on ∂Ω, due
to the nonlocal character of the operator (−∆)
s
.
In what follows we will denote by e
k, A
s
and e
k, s
, k ∈ N , the k–th eigenfunction of A
s
and (−∆)
s
, respectively.
Taking into account the definition of A
s
, it is easily seen that its eigenfunctions e
k, A
s
,
k ∈ N , are exactly the eigenfunctions of the Laplace operator −∆, i.e.
e
k, A
s
= e
k
.
Also, since e
k
∈ C
∞
(Ω) ∩ C
m
(Ω) for any m ∈ N (see, for instance, [11]), then
(2.3) e
k, A
s
∈ C
∞
(Ω) ∩ C
m
(Ω) .
Of course, constructing the eigenfunctions of (−∆)
s
is more difficult. In spite of this, we
have some regularity results for them. Precisely, denoting by δ(x) = dist(x, ∂Ω), x ∈ R
n
,
by [22, Theorems 1.1 and 1.3] and [30, Proposition 4], we have that
e
k, s
/δ
s
|Ω
∈ C
0,α
(Ω) for some α ∈ (0, 1) ,
namely e
k, s
/δ
s
|Ω
has a continuous extension to Ω which is C
0,α
(Ω) . In particular, e
k, s
is
H¨older continuous up to the boundary.
Aim of this section will be to show that the H¨older regularity is optimal for the eigen-
functions e
k, s
of (−∆)
s
. To this purpose, first of all we recall the notion of Poisson kernel
of fractional type and, then, we discuss the optimal regularity of the eigenfunctions e
k, s
.
4 R. SERVADEI AND E. VALDINOCI
2.1. Poisson kernel of fractional type. Here we recall the notion of Poisson kernels of
fractional type and their relation with the Dirichlet problem (see [20, Chapter I]).
First of all, for any r > 0, x ∈ B
r
(that is the ball of radius r centered at the origin)
and y ∈ R
n
\ B
r
, we define
P
r
(x, y) := c
o
(n, s)
r
2
− |x|
2
|y|
2
− r
2
s
1
|x −y|
n
,
with c
o
(n, s) > 0. It is known (see [20, Appendix]) that, for any fixed x ∈ B
r
the function
I(x) :=
Z
R
n
\B
r
P
r
(x, y) dy
is constant in x. Therefore, we normalize c
o
(n, s) in such a way that
3
(2.4)
Z
R
n
\B
r
P
r
(x, y) dy = 1 .
The function P
r
plays the role of a fractional Poisson kernel, namely if g ∈ C(R
n
)∩L
∞
(R
n
)
and
(2.5) u
g
(x) :=
Z
R
n
\B
r
P
r
(x, y) g(y) dy if x ∈ B
r
g(x) if x ∈ R
n
\ B
r
,
then u
g
is the unique solution of
(2.6)
(−∆)
s
u
g
= 0 in B
r
u
g
= g outside B
r
.
For this, see [20, 33].
2.2. Optimal regularity for the eigenfunctions of (−∆)
s
. In this subsection we prove
that the C
0, α
-regularity of the eigenfunctions e
k, s
is optimal. Precisely, we show that, in
general, the eigenfunctions of (−∆)
s
need not to be Lipschitz continuous up to the boundary
(i.e. the H¨older regularity is optimal).
For concreteness, we consider the case
(2.7) n > 2s ,
the domain Ω := B
r
and the first eigenfunction e
1, s
(normalized in such a way that
ke
1, s
k
L
2
(R
n
)
= 1 and e
1, s
> 0 in R
n
, see [28, Proposition 9 and Appendix A]) of (−∆)
s
in
B
r
, i.e.
(2.8)
(−∆)
s
e
1, s
= λ
1, s
e
1, s
in B
r
e
1, s
= 0 in R
n
\ B
r
.
We prove that
Proposition 2. The function e
1, s
given in (2.8) is such that
e
1, s
6∈ W
1,∞
(B
r
) .
Proof. The proof is by contradiction. We suppose that e
1, s
∈ W
1,∞
(B
r
) and so e
1, s
∈
W
1,∞
(R
n
), that is
(2.9) |e
1, s
(x)| + |∇e
1, s
(x)| 6 M, x ∈ R
n
for some M > 0.
From now on, we proceed by steps.
Step 1. The function e
1, s
is spherically symmetric and radially decreasing in R
n
.
3
More explicitly, one can choose c
o
(n, s) := Γ(n/2) sin(πs)/π
(n/2)+1
, see [20, pages 399–400].
ON THE SPECTRUM OF TWO DIFFERENT FRACTIONAL OPERATORS 5
Proof. For this, since e
1, s
> 0 in R
n
, we consider its symmetric radially decreasing re-
arrangement e
?
1, s
(see, e.g., [19, Chapter 2] for the basics of such a rearrangement). We
observe that e
?
1, s
vanishes outside B
r
, since so does e
1, s
. Moreover, we recall that the L
2
-
norm is preserved by the rearrangement, while the fractional Gagliardo seminorm decreases,
see, e.g. [1, 3, 21]. Then, by this and since λ
1, s
is obtained by minimizing the fractional
Gagliardo seminorm under constraint on the L
2
-norm for functions that vanish outside B
r
(see [28, Proposition 9]), we conclude that the minimum is attained by e
?
1, s
(as well as by
e
1, s
).
Since λ
1, s
is a simple eigenvalue (see [28, Proposition 9 and Appendix A]), it follows that
e
?
1, s
= e
1, s
and Step 1 is proved.
Now, let Q be the fractional fundamental solution given by
Q(x) := c
1
(n, s)|x|
2s−n
, x ∈ R
n
\ {0}.
Here the constant c
1
(n, s) > 0 is chosen in such a way that (−∆)
s
Q is the Dirac’s delta δ
0
centered at the origin (see, e.g., [20, page 44] for the basic properties of fractional funda-
mental solutions).
We define
(2.10) ˜v(x) := λ
1, s
Q ∗ e
1, s
(x) = λ
1, s
c
1
(n, s)
Z
R
n
|y|
2s−n
e
1, s
(x − y) dy , x ∈ R
n
and
(2.11) v(x) := e
1, s
(x) − ˜v(x) , x ∈ R
n
.
First of all, notice that ˜v > 0 in R
n
, since λ
1, s
> 0 , Q > 0 and e
1, s
> 0 in R
n
.
Step 2. The function ˜v is spherically symmetric and radially decreasing in R
n
.
Proof. Indeed, if R is a rotation, we use Step 1 and the substitution ˜y := Ry to obtain for
any x ∈ R
n
˜v(x) = λ
1, s
c
1
(n, s)
Z
R
n
|y|
2s−n
e
1, s
(x − y) dy =
= λ
1, s
c
1
(n, s)
Z
R
n
|y|
2s−n
e
1, s
R(x −y)
dy
= λ
1, s
c
1
(n, s)
Z
R
n
|˜y|
2s−n
e
1, s
(Rx − ˜y) d˜y = ˜v(Rx),
that shows the spherical symmetry of ˜v.
As for the fact that ˜v is radially decreasing in R
n
, we take ρ > 0 and define
(2.12) v
?
(ρ) := −
λ
1, s
c
1
(n, s)
−1
˜v(0, . . . , 0, ρ) = −
Z
R
n
|y|
2s−n
e
1, s
(−y
0
, ρ − y
n
) dy,
where we used the notation y = (y
0
, y
n
) ∈ R
n−1
× R for the coordinates in R
n
.
The goal is to show that for any ρ > 0
(2.13) v
0
?
(ρ) > 0.
For this, first note that
v
?
(ρ) = −
Z
R
n
∩{|ρ−y
n
|6r}
|y|
2s−n
e
1, s
(−y
0
, ρ − y
n
) dy
−
Z
R
n
∩{|ρ−y
n
|>r}
|y|
2s−n
e
1, s
(−y
0
, ρ − y
n
) dy
= −
Z
R
n
∩{|ρ−y
n
|6r}
|y|
2s−n
e
1, s
(−y
0
, ρ − y
n
) dy ,