1 Introduction
Let us start with a brief description of the classical heat content for the heat equation.In [12] (see also [13]) the heat content of a Borel measurable set D⊂ℝN at time t is defined as
ℍ
D
(
t
)
:=
∫
D
T
(
t
)
χ
D
(
x
)
𝑑
x
,
with (T(t))t≥0 being the heat semigroup in L2(ℝN).Therefore, ℍD(t) represents the amount of heat in D at time t if in D the initial temperature is 1 and in ℝN∖D the initial temperature is 0. The following characterization for the perimeter of a set D⊂ℝN with finite perimeter was given in [10, Theorem 3.3]:
(1.1)
lim
t
→
0
+
π
t
∫
ℝ
N
∖
D
T
(
t
)
χ
D
(
x
)
𝑑
x
=
P
(
D
)
.
As a consequence, the following result was presented in [12] for the heat content: for an open subset D in ℝN with finite Lebesgue measure and finite perimeter there holds
(1.2)
ℍ
D
(
t
)
=
|
D
|
-
t
π
P
(
D
)
+
o
(
t
)
as
t
↓
0
.
In [1] and [2], the concept of heat content was extended to more general diffusion processes. More precisely, for 0<α≤2 let pt(α):ℝN→[0,∞) be the probability density such that its Fourier transform verifies
p
t
(
α
)
^
(
x
)
=
e
-
t
|
x
|
α
.
If one considers
T
t
(
α
)
(
f
)
(
x
)
:=
∫
ℝ
N
f
(
y
)
p
t
(
α
)
(
x
-
y
)
𝑑
y
,
thenu(x,t)=Tt(α)(f)(x) is the unique weak solution of the initial valued problem
{
u
t
(
x
,
t
)
=
-
(
-
Δ
)
α
2
u
(
x
,
t
)
,
(
x
,
t
)
∈
ℝ
N
×
[
0
,
∞
)
,
u
(
x
,
0
)
=
f
(
x
)
,
x
∈
ℝ
N
.
And, in this context, the heat content of a Borel measurable set D⊂ℝN at time t is defined as
ℍ
D
(
α
)
(
t
)
:=
∫
D
T
t
(
α
)
χ
D
(
x
)
𝑑
x
.
Note that for α=2,
p
t
(
2
)
=
(
4
π
t
)
-
N
2
exp
(
-
|
x
|
2
4
t
)
is the Gaussian kerneland consequently ℍD(2)(t)=ℍD(t);while forα=1 and kN=Γ(N+12)/π(N+1)/2,
p
t
(
1
)
(
x
)
=
k
N
t
(
t
+
|
x
|
)
N
+
1
2
is the Poisson heat kernel.
In recent years, nonlocal diffusion problems with non-singular kernels and related problems have also been studied; see for example [3] and the references therein. More concretely, for a measurable, nonnegative and radially symmetricfunction J:ℝN→[0,+∞) verifying ∫ℝNJ(z)𝑑z=1, the following nonlocal Cauchy problem has been studied in [3]:
(1.3)
{
u
t
(
x
,
t
)
=
∫
ℝ
N
J
(
x
-
y
)
(
u
(
y
,
t
)
-
u
(
x
,
t
)
)
𝑑
y
,
(
x
,
t
)
∈
ℝ
N
×
[
0
,
+
∞
)
,
u
(
x
,
0
)
=
u
0
(
x
)
,
x
∈
ℝ
N
.
This equation appears naturally from the following considerations: ifu(x,t) is thought of as a density at a point x at time t, andJ(x-y) is thought of as the probability distribution of jumpingfrom location y to location x, then ∫ℝNJ(y-x)u(y,t)𝑑y=(J∗u)(x,t) is the rate at which individualsare arriving at position x from all other places, and -u(x,t)=-∫ℝNJ(y-x)u(x,t)𝑑y is the rate at which they areleaving location x to travel to all other sites. Thisconsideration, in the absence of external or internal sources,leads immediately to the fact that the density u satisfiesequation (1.3).
Recall from [3] that a solution of (1.3) in the time interval [0,T] is a function u∈W1,1(0,T;L2(ℝN)) which satisfies u(x,0)=u0 and
u
t
(
x
,
t
)
=
∫
ℝ
N
J
(
x
-
y
)
(
u
(
y
,
t
)
-
u
(
x
,
t
)
)
𝑑
y
a.e. in
(
x
,
t
)
∈
ℝ
N
×
(
0
,
T
)
.
A simple integration of the equation in (1.3) in space gives that the total mass is preserved, that is,
(1.4)
∫
ℝ
N
u
(
x
,
t
)
𝑑
x
=
∫
ℝ
N
u
0
(
x
)
𝑑
x
for all
t
≥
0
.
Our aim is to study the heat content associated with the above nonlocal diffusion processes, and to relate it with the (local) heat content.
Let us fix the hypothesis on the kernel J that we will assume in this paper:
J
∈
𝒞
(
ℝ
N
,
ℝ
)
is non-negative, radially decreasing and compactly supported with ∫ℝNJ(x)𝑑x=1.
Definition 1.1
Given a Lebesgue measurable set D⊂RN with finite measure, we define the J-heat content of D in RN at time t by
ℍ
D
J
(
t
)
:=
∫
D
u
(
x
,
t
)
𝑑
x
,
with u being the solution of (1.3) with the datum u0=χD.
Note that, from (1.4), we have
ℍ
D
J
(
0
)
=
|
D
|
and that no further regularity is required for D besides having finite measure.
We have the following interpretation of the J-heat content ℍDJ(t): if u(x,t) represents the density of a population at a point x∈ℝN at time t with initial conditionu(x,0)=χD(x), then the J-heat content of D at time t represents the size of the population that remained inside D at that time when in D the initial density of the population is 1 and in ℝN∖D the initial density of the population is 0.
Our first result gives an asymptotic expansion of the J-heat content, from which a result similar to (1.2) follows. This expansion, obtained from a classical Taylor’s expansion, is given for any time and for any order and with the main fact that all the terms involved in the expansion can be expressed using nonlocal perimeters of the set D with different kernels.We will use the following notation:
J
=
(
J
∗
)
1
,
J
∗
J
=
(
J
∗
)
2
,
and for the convolution of n kernels:
J
∗
J
∗
⋯
∗
J
=
(
J
∗
)
n
.
Observe that, for all n, the term (J∗)n is non-negative, radially decreasingand compactly supported with
∫
ℝ
N
(
J
∗
)
n
(
x
)
d
x
=
1
.
One of our main results is the following.
Theorem 1.2
For the J-heat contentHDJ(t) of D given in Definition 1.1 there holds
(1.5)
ℍ
D
J
(
t
)
=
|
D
|
-
∑
n
=
1
+
∞
(
∑
k
=
1
n
(
n
k
)
(
-
1
)
n
-
k
P
(
J
∗
)
k
(
D
)
)
t
n
n
!
for all
t
>
0
,
where P(J∗)k(D) denotes the (J∗)k-nonlocal perimeter given by
P
(
J
∗
)
k
(
D
)
:=
∫
D
∫
ℝ
N
∖
D
(
J
∗
)
k
(
x
-
y
)
d
y
d
x
.
Moreover, for L≥1 we have
(1.6)
|
ℍ
D
J
(
t
)
-
|
D
|
+
∑
n
=
1
L
(
∑
k
=
1
n
(
n
k
)
(
-
1
)
n
-
k
P
(
J
∗
)
k
(
D
)
)
t
n
n
!
|
≤
|
D
|
2
L
(
L
+
1
)
!
t
L
+
1
for all
t
>
0
.
We also have the following expression:
(1.7)
ℍ
D
J
(
t
)
=
∑
n
=
0
+
∞
(
∫
D
∫
D
(
J
∗
)
n
(
x
-
y
)
d
y
d
x
)
e
-
t
t
n
n
!
for all
t
>
0
,
where∫D∫D(J∗)0(x-y)dydx=|D|.
The J-nonlocal perimeter of a measurable set was studied in [9] for non-singular kernels in connection with other geometrical properties.Here we obtain, in particular, the following nonlocal version of (1.1) and (1.2): let u be the solution of (1.3) for the datum u0=χD, with a finite Lebesgue measurable set D in ℝN; then
(1.8)
lim
t
→
0
+
1
t
∫
ℝ
N
∖
D
u
(
x
,
t
)
𝑑
x
=
-
(
ℍ
D
J
)
′
(
0
)
=
P
J
(
D
)
,
or equivalently
(1.9)
ℍ
D
J
(
t
)
=
|
D
|
-
P
J
(
D
)
t
+
o
(
t
)
as
t
↓
0
.
Remark 1.3
(i) There is a remarkable point concerning (1.9) and (1.2): from the small asymptotic expansion of ℍDJ(t) and ℍD(t), it is possible to recover a geometric feature of the set D in addition to its volume, namely, we can obtain the J-perimeter and the classical perimeter, respectively.
(ii) The result given in (1.8) is similar to the one obtained in [2] for ℍD(α)(t), 0<α<1, which says that
(
ℍ
D
(
α
)
)
′
(
0
)
=
-
𝒜
α
,
N
𝒫
α
(
D
)
,
where 𝒜α,N is a positive constant and 𝒫α(D) is the α-perimeter
𝒫
α
(
D
)
:=
∫
D
∫
ℝ
N
∖
D
d
x
d
y
|
x
-
y
|
N
+
α
.
We also show that we can recover the classical heat content from the J-heat content rescaling the kernel.Let
J
ε
(
x
)
:=
1
ε
N
J
(
x
ε
)
.
Associated with the rescaled kernel Jε we can consider the Jε-heat content justby taking u as the solution to (1.3) with J=Jε in Definition 1.1.
Theorem 1.4
For a subset D in RN with finite Lebesgue measure we have
lim
ε
→
0
+
ℍ
D
J
ε
(
C
J
ε
2
t
)
=
ℍ
D
(
t
)
for all
t
>
0
.
Here CJ is the normalizing constant given by
C
J
=
2
∫
ℝ
N
J
(
x
)
|
x
N
|
2
𝑑
x
.
Let us now consider the spectral heat content given by
ℚ
D
(
t
)
:=
∫
D
u
(
x
,
t
)
𝑑
x
for the solution u of the Dirichlet problem
{
u
t
(
x
,
t
)
=
Δ
u
(
x
,
t
)
,
(
x
,
t
)
∈
D
×
[
0
,
∞
)
,
u
(
x
,
t
)
=
0
,
(
x
,
t
)
∈
∂
D
×
(
0
,
∞
)
,
u
(
x
,
0
)
=
χ
D
(
x
)
,
x
∈
D
.
The following result was given in [14] for smooth bounded domains:
ℚ
D
(
t
)
=
|
D
|
-
2
π
P
(
D
)
t
+
1
2
(
N
-
1
)
∫
∂
D
ℋ
∂
D
(
x
)
𝑑
ℋ
N
-
1
(
x
)
t
+
O
(
t
3
/
2
)
as
t
↓
0
,
where ℋ∂D is the mean curvature of ∂D.
In [3], the following nonlocal Dirichlet problem has also been studied:
(1.10)
{
u
t
(
x
,
t
)
=
∫
ℝ
N
J
(
x
-
y
)
(
u
(
y
,
t
)
-
u
(
x
,
t
)
)
𝑑
y
,
(
x
,
t
)
∈
D
×
[
0
,
∞
)
,
u
(
x
,
t
)
=
0
,
(
x
,
t
)
∈
(
ℝ
N
∖
D
)
×
[
0
,
∞
)
,
u
(
x
,
0
)
=
u
0
(
x
)
,
x
∈
ℝ
N
.
Therefore, we can also define
ℚ
D
J
(
t
)
:=
∫
D
u
(
x
,
t
)
𝑑
x
,
with u being the solution of (1.10) for the datum u0=χD. Observe that also
ℚ
D
J
(
0
)
=
|
D
|
.
We prove the following result.
Theorem 1.5
For the J-spectral heat contentQDJ(t) of D it holds that
ℚ
D
J
(
t
)
=
|
D
|
-
P
J
(
D
)
t
+
1
2
∫
D
ℋ
∂
D
J
(
x
)
𝑑
x
t
2
+
1
2
∫
D
∫
D
∫
D
J
(
x
-
y
)
J
(
y
-
z
)
𝑑
z
𝑑
y
𝑑
x
t
2
+
O
(
t
3
)
as
t
↓
0
,
where H∂DJ(x) is the J-mean curvature at x defined by
ℋ
∂
D
J
(
x
)
:=
∫
ℝ
N
J
(
x
-
y
)
(
χ
ℝ
N
∖
D
(
y
)
-
χ
D
(
y
)
)
𝑑
y
.
2 The J-Heat Content: Proofs of the Results
We will denote by ΔJ the nonlocal operator of equation (1.3), i.e., we let
Δ
J
u
(
x
)
:=
∫
ℝ
N
J
(
x
-
y
)
(
u
(
y
)
-
u
(
x
)
)
𝑑
y
.
In [3], it is showed that the operator B in L2(Ω) with domain D(B)=L2(Ω) defined by
B
(
u
)
=
v
⇔
v
(
x
)
=
-
Δ
J
u
(
x
)
for all
x
∈
Ω
is m-completely accretive in L2(Ω). Then this operator B generates a C0-semigroup (TJ(t))t≥0 in L2(Ω) which solves the Cauchy problem (1.3). With this operator we can describe the J-heat content of a bounded subset D⊂ℝN as
ℍ
D
J
(
t
)
=
∫
D
T
J
(
t
)
χ
D
(
x
)
𝑑
x
.
Proposition 2.1
Given a Lebesgue measurable set D⊂RN with finite measure, we have
ℍ
D
J
(
t
)
=
∥
T
J
(
t
2
)
χ
D
∥
L
2
2
.
Proof.
It is enough to prove that the operators TJ(t) are selfadjoint since then
∥
T
J
(
t
2
)
χ
D
∥
L
2
2
=
〈
T
J
(
t
2
)
χ
D
,
T
J
(
t
2
)
χ
D
〉
=
〈
T
J
(
t
)
χ
D
,
χ
D
〉
=
ℍ
D
J
(
t
)
.
So, let us show that TJ(t) is selfadjoint. Let ℱ:L2(ℝN)→L2(ℝN) be the Fourier–Plancherel transform. We write f^:=ℱ(f). If u(t)(x)=u(x,t):=TJ(t)f(x), since ut(t)=J*u(t)-u(t), applying the Fourier–Plancherel transform, we have
(2.1)
u
^
t
(
ξ
,
t
)
=
(
J
^
-
1
)
u
^
(
ξ
,
t
)
,
from which it follows that
u
^
(
ξ
,
t
)
=
e
(
J
^
(
ξ
)
-
1
)
t
f
^
(
ξ
)
.
Therefore, given f,g∈L2(ℝN), we have
〈
T
J
(
t
)
f
,
g
〉
=
〈
ℱ
(
T
J
(
t
)
f
)
,
ℱ
(
g
)
〉
=
〈
e
(
J
^
(
ξ
)
-
1
)
t
f
^
,
g
^
〉
=
〈
f
^
,
e
(
J
^
(
ξ
)
-
1
)
t
g
^
〉
=
〈
f
,
T
J
(
t
)
g
〉
,
as we wanted to show.∎
Associated with the non-singular kernel J there is a nonlocal version of the usual perimeter of a set and a nonlocal concept of mean curvature (see [5], [9]): let E⊂ℝN be a measurable set; the nonlocal J-perimeter ofE is given by
P
J
(
E
)
:=
∫
E
(
∫
ℝ
N
∖
E
J
(
x
-
y
)
𝑑
y
)
𝑑
x
,
and the J-mean curvature at a point x is defined by
ℋ
∂
E
J
(
x
)
:=
∫
ℝ
N
J
(
x
-
y
)
(
χ
ℝ
N
∖
E
(
y
)
-
χ
E
(
y
)
)
𝑑
y
.
Observe that, since ∫ℝNJ(x)𝑑x=1, there holds
|
E
|
=
∫
ℝ
N
∫
E
J
(
x
-
y
)
𝑑
y
𝑑
x
=
∫
E
∫
E
J
(
x
-
y
)
𝑑
y
𝑑
x
+
∫
ℝ
N
∖
E
∫
E
J
(
x
-
y
)
𝑑
y
𝑑
x
,
that is,
(2.2)
|
E
|
=
∫
E
∫
E
J
(
x
-
y
)
𝑑
y
𝑑
x
+
P
J
(
E
)
,
and also
∫
E
ℋ
∂
E
J
(
x
)
𝑑
x
=
∫
E
(
1
-
2
∫
E
J
(
x
-
y
)
𝑑
y
)
𝑑
x
=
|
E
|
-
2
∫
E
∫
E
J
(
x
-
y
)
𝑑
y
𝑑
x
,
from which
(2.3)
∫
E
ℋ
∂
E
J
(
x
)
𝑑
x
=
2
P
J
(
E
)
-
|
E
|
follows on account of (2.2).
2.1 The Asymptotic Expansion of the J-Heat Content
Proof of Theorem 1.2.
From (2.1) we have that for u(⋅,t)=TJ(t)χD the Fourier transform verifiesthe evolution problem
u
^
t
(
ξ
,
t
)
=
(
J
^
(
ξ
)
-
1
)
u
^
(
ξ
,
t
)
,
with the initial conditionu^(ξ,0)=χD^(ξ).Hence,
(2.4)
u
^
(
ξ
,
t
)
=
e
(
J
^
(
ξ
)
-
1
)
t
χ
D
^
(
ξ
)
.
Using Taylor’s expansion of the exponential, we deduce
u
^
(
ξ
,
t
)
=
∑
n
=
0
L
(
J
^
(
ξ
)
-
1
)
n
χ
D
^
(
ξ
)
t
n
n
!
+
(
J
^
(
ξ
)
-
1
)
L
+
1
e
(
J
^
(
ξ
)
-
1
)
s
χ
D
^
(
ξ
)
t
L
+
1
(
L
+
1
)
!
=
∑
n
=
0
L
(
χ
D
^
(
ξ
)
+
∑
k
=
1
n
(
n
k
)
(
-
1
)
n
-
k
J
^
(
ξ
)
k
χ
D
^
(
ξ
)
)
t
n
n
!
+
(
J
^
(
ξ
)
-
1
)
L
+
1
e
(
J
^
(
ξ
)
-
1
)
s
χ
D
^
(
ξ
)
t
L
+
1
(
L
+
1
)
!
=
∑
n
=
0
L
(
χ
D
^
(
ξ
)
+
∑
k
=
1
n
(
n
k
)
(
-
1
)
n
-
k
J
^
(
ξ
)
k
χ
D
^
(
ξ
)
)
t
n
n
!
+
(
J
^
(
ξ
)
-
1
)
L
+
1
u
^
(
ξ
,
s
)
t
L
+
1
(
L
+
1
)
!
for 0<s<t.Taking now the inverse Fourier–Plancherel transform and integrating over D, we get
ℍ
D
J
(
t
)
=
∑
n
=
0
L
(
(
-
1
)
n
|
D
|
+
∑
k
=
1
n
(
n
k
)
(
-
1
)
n
-
k
∫
D
∫
D
(
J
∗
)
k
(
x
-
y
)
d
y
d
x
)
t
n
n
!
+
∫
D
ℱ
-
1
(
(
J
^
(
ξ
)
-
1
)
L
+
1
u
^
(
ξ
,
s
)
)
(
x
)
𝑑
x
t
L
+
1
(
L
+
1
)
!
for n≥1.Now, for n≥1, using (2.2) yields
(
-
1
)
n
|
D
|
+
∑
k
=
1
n
(
n
k
)
(
-
1
)
n
-
k
∫
D
∫
D
(
J
∗
)
k
(
x
-
y
)
d
y
d
x
=
(
-
1
)
n
|
D
|
+
∑
k
=
1
n
(
n
k
)
(
-
1
)
n
-
k
(
|
D
|
-
P
(
J
∗
)
k
(
D
)
)
=
-
∑
k
=
1
n
(
n
k
)
(
-
1
)
n
-
k
P
(
J
∗
)
k
(
D
)
.
Also,
(2.5)
∫
D
ℱ
-
1
(
(
J
^
(
ξ
)
-
1
)
L
+
1
u
^
(
ξ
,
s
)
)
(
x
)
d
x
=
(
-
1
)
L
+
1
|
D
|
+
∑
k
=
1
L
+
1
(
L
+
1
k
)
(
-
1
)
L
+
1
-
k
∫
D
(
J
∗
)
k
∗
u
(
x
,
s
)
d
x
.
Using the fact that for this problem we have 0≤u≤1(this follows from the maximum principle),we get
0
≤
∫
D
(
(
J
∗
)
k
∗
u
)
(
x
,
t
)
d
x
≤
|
D
|
for all
k
∈
ℕ
.
Hence, from (2.5) we have that
|
∫
D
ℱ
-
1
(
(
J
^
(
ξ
)
-
1
)
L
+
1
u
^
(
ξ
,
s
)
)
(
x
)
𝑑
x
|
≤
1
2
∑
k
=
0
L
+
1
(
L
+
1
k
)
|
D
|
=
2
L
|
D
|
.
From which (1.5) and (1.6) follow.
From (2.4) we also have
u
^
(
ξ
,
t
)
=
e
-
t
χ
D
^
(
ξ
)
e
t
J
^
(
ξ
)
=
e
-
t
∑
n
=
0
∞
(
J
^
(
ξ
)
)
n
χ
D
^
(
ξ
)
t
n
n
!
.
Taking here the inverse Fourier transform, we get that
(2.6)
u
(
x
,
t
)
=
e
-
t
∑
n
=
0
∞
ℱ
-
1
(
(
J
^
(
ξ
)
)
n
χ
D
^
(
ξ
)
)
(
x
,
t
)
t
n
n
!
=
∑
n
=
0
∞
∫
D
(
J
∗
)
n
(
x
-
y
)
d
y
e
-
t
t
n
n
!
,
where ∫D(J∗)0(x-y)dy=χD(x). Then, integrating over D, we get
ℍ
D
J
(
t
)
=
∑
n
=
0
∞
∫
D
∫
D
(
J
∗
)
n
(
x
-
y
)
d
y
d
x
e
-
t
t
n
n
!
,
and thus (1.7) is proved.∎
Remark 2.2
(i) We can also proceed in the following way: Integrating in (1.3) over D, we get
(
ℍ
D
J
)
′
(
t
)
=
∫
D
∫
ℝ
N
J
(
x
-
y
)
u
(
y
,
t
)
𝑑
y
𝑑
x
-
∫
D
u
(
x
,
t
)
𝑑
x
=
∫
D
∫
ℝ
N
J
(
x
-
y
)
u
(
y
,
t
)
𝑑
y
𝑑
x
-
ℍ
D
J
(
t
)
,
and hence
(2.7)
ℍ
D
J
(
t
)
+
(
ℍ
D
J
)
′
(
t
)
=
∫
D
∫
ℝ
N
J
(
x
-
y
)
u
(
y
,
t
)
𝑑
y
𝑑
x
.
Taking the time derivative in (2.7) and using again (2.7), we get
(
ℍ
D
J
)
′
(
t
)
+
(
ℍ
D
J
)
′′
(
t
)
=
∫
D
∫
ℝ
N
J
(
x
-
y
)
u
t
(
y
,
t
)
𝑑
y
𝑑
x
=
∫
D
∫
ℝ
N
J
(
x
-
y
)
(
∫
ℝ
N
(
J
(
y
-
z
)
u
(
z
,
t
)
-
u
(
y
,
t
)
)
𝑑
x
)
𝑑
y
𝑑
x
=
∫
D
∫
ℝ
N
∫
ℝ
N
J
(
x
-
y
)
J
(
y
-
z
)
u
(
z
,
t
)
𝑑
z
𝑑
y
𝑑
x
-
∫
D
∫
ℝ
N
J
(
x
-
y
)
u
(
y
,
t
)
𝑑
y
𝑑
x
=
∫
D
(
(
J
∗
)
2
∗
u
)
(
x
,
t
)
d
x
-
ℍ
D
J
(
t
)
-
(
ℍ
D
J
)
′
(
t
)
,
hence
ℍ
D
J
(
t
)
+
2
(
ℍ
D
J
)
′
(
t
)
+
(
ℍ
D
J
)
′′
(
t
)
=
∫
D
(
(
J
∗
)
2
∗
u
)
(
x
,
t
)
d
x
.
By induction, it is easy to see that
∑
k
=
0
n
(
n
k
)
(
ℍ
D
J
)
(
k
)
(
t
)
=
∫
D
(
(
J
∗
)
n
∗
u
)
(
x
,
t
)
d
x
for any n≥1,which is equivalent to what was obtained before.
(ii) In particular, we have
(2.8)
ℍ
D
J
(
0
)
+
2
(
ℍ
D
J
)
′
(
0
)
+
(
ℍ
D
J
)
′′
(
0
)
=
∫
D
∫
ℝ
N
∫
D
J
(
x
-
y
)
J
(
y
-
z
)
𝑑
z
𝑑
y
𝑑
x
.
Then, having in mind (2.3), we get
(2.9)
(
ℍ
D
J
)
′′
(
0
)
=
∫
D
∫
ℝ
N
∫
D
J
(
x
-
y
)
J
(
y
-
z
)
𝑑
z
𝑑
y
𝑑
x
+
∫
D
ℋ
∂
D
J
(
x
)
𝑑
x
.
Therefore, applying Taylor’s expansion, we obtain
ℍ
D
J
(
t
)
=
|
D
|
-
P
J
(
D
)
t
+
1
2
∫
D
ℋ
∂
D
J
(
x
)
𝑑
x
t
2
+
1
2
∫
D
∫
ℝ
N
∫
D
J
(
x
-
y
)
J
(
y
-
z
)
𝑑
z
𝑑
y
𝑑
x
t
2
+
O
(
t
3
)
as
t
↓
0
.
(iii) For n≥2 we can express the coefficients (ℍDJ)(n)(0) by using the nonlocal curvature as follows:
(2.10)
(
ℍ
D
J
)
(
n
)
(
0
)
=
(
-
1
)
n
P
J
(
D
)
+
1
2
∑
k
=
1
n
-
1
(
n
-
1
k
)
(
-
1
)
n
-
1
-
k
(
∫
D
ℋ
∂
D
(
J
∗
)
k
(
x
)
𝑑
x
-
∫
D
ℋ
∂
D
(
J
∗
)
(
k
+
1
)
(
x
)
𝑑
x
)
.
Indeed,
(
ℍ
D
J
)
(
n
)
(
0
)
=
-
∑
k
=
1
n
(
n
k
)
(
-
1
)
n
-
k
P
(
J
∗
)
k
(
D
)
=
∑
k
=
1
n
-
1
(
-
1
)
n
-
1
-
k
(
(
n
-
1
k
-
1
)
+
(
n
-
1
k
)
)
P
(
J
∗
)
k
(
D
)
-
P
(
J
∗
)
n
(
D
)
=
(
-
1
)
n
P
J
(
D
)
+
∑
k
=
1
n
-
1
(
n
-
1
k
)
(
-
1
)
n
-
1
-
k
(
P
(
J
∗
)
k
(
D
)
-
P
(
J
∗
)
(
k
+
1
)
(
D
)
)
,
and hence, using (2.3), we can write
P
(
J
∗
)
k
(
D
)
-
P
(
J
∗
)
(
k
+
1
)
(
D
)
=
1
2
(
∫
D
ℋ
∂
D
(
J
∗
)
k
(
x
)
𝑑
x
-
∫
D
ℋ
∂
D
(
J
∗
)
(
k
+
1
)
(
x
)
𝑑
x
)
to get (2.10).
Remark 2.3
Now, let us make a comment on the relation of the semigroup TJ(t) and the operator ΔJχD.
Observe that for ϕ(x,t):=TJ(t)χD(x)-χD(x), by (1.4) we have
∫
ℝ
N
ϕ
(
x
,
t
)
𝑑
x
=
0
for all
t
≥
0
,
hence
∫
ℝ
N
ϕ
+
(
x
,
t
)
𝑑
x
=
∫
ℝ
N
ϕ
-
(
x
,
t
)
𝑑
x
for all
t
≥
0
.
Therefore, since
ϕ
+
(
x
,
t
)
=
(
T
J
(
t
)
χ
D
(
x
)
-
χ
D
(
x
)
)
χ
ℝ
N
∖
D
(
x
)
,
we have
∥
T
J
(
t
)
χ
D
-
χ
D
∥
L
1
=
2
∫
ℝ
N
ϕ
+
(
x
,
t
)
𝑑
x
=
2
∫
ℝ
N
∖
D
T
J
(
t
)
χ
D
(
x
)
𝑑
x
.
Then, by (1.8), we get
(2.11)
lim
t
→
0
+
1
2
t
∥
T
J
(
t
)
χ
D
-
χ
D
∥
L
1
=
P
J
(
D
)
.
Note that this formula is similar to the one that holds for the classical heat content; see [10].
Now, since
P
J
(
D
)
=
1
2
∫
ℝ
N
∫
ℝ
N
J
(
x
-
y
)
|
χ
D
(
y
)
-
χ
D
(
x
)
|
𝑑
y
𝑑
x
=
1
2
∫
ℝ
N
|
∫
ℝ
N
J
(
x
-
y
)
(
χ
D
(
y
)
-
χ
D
(
x
)
)
𝑑
y
|
𝑑
x
=
1
2
∥
Δ
J
χ
D
∥
L
1
,
we can write (2.11) as
(2.12)
lim
t
→
0
+
1
t
∥
T
J
(
t
)
χ
D
-
χ
D
∥
L
1
=
∥
Δ
J
χ
D
∥
L
1
.
Moreover, since the operator B=ΔJ is the infinitesimal generator of the C0-semigroup (TJ(t))t≥0 in L2(ℝN), we have
(2.13)
Δ
J
χ
D
=
lim
t
→
0
+
1
t
(
T
J
(
t
)
χ
D
-
χ
D
)
in
L
2
(
ℝ
N
)
.
Hence, (2.12) and (2.13) imply that also
Δ
J
χ
D
=
lim
t
→
0
+
1
t
(
T
J
(
t
)
χ
D
-
χ
D
)
in
L
1
(
ℝ
N
)
.
2.2 A Probabilistic Interpretation
Let us show that the formula given in (1.7), that is,
ℍ
D
J
(
t
)
=
∑
k
=
0
+
∞
(
∫
D
∫
D
(
J
∗
)
k
(
x
-
y
)
d
y
d
x
)
e
-
t
t
k
k
!
,
has a probabilistic interpretation.
As mentioned in Section 1, J(x-y) is thought of as the probability distribution of jumpingfrom location y to location x. Then
∫
ℝ
N
J
(
x
-
s
)
J
(
s
-
y
)
𝑑
s
is the probability of jumpingfrom location y to location x in two jumps (the probability of passing through a point s is J(x-s)J(s-y) and we integrate for s∈ℝN). Further,
∫
ℝ
N
∫
ℝ
N
J
(
x
-
s
)
J
(
s
-
w
)
J
(
w
-
y
)
𝑑
s
𝑑
w
is the probability of jumpingfrom location y to location x in three jumps.
Then from J(x-y) we obtain the probability of a transition from x to D in k steps as
F
(
k
)
(
x
,
D
)
=
∫
D
(
J
∗
)
k
(
x
-
y
)
d
y
,
and we can also set
F
(
0
)
(
x
,
D
)
=
χ
D
(
x
)
.
Observe that F(k)(x,D)also determines how much matter of D goes to x after k jumps, even for k=0.Furthermore, if we define
f
(
0
)
:=
∫
D
∫
D
(
J
∗
)
0
(
x
-
y
)
d
y
d
x
=
|
D
|
and
f
(
k
)
:=
∫
D
∫
D
(
J
∗
)
k
(
x
-
y
)
d
y
d
x
,
k
=
1
,
2
,
…
,
we have that f(k) is the amount of matter of D remaining in D after k jumps for any k≥0. Let us call these jumps J-jumps.
From (2.6) we have that u(x,t) is the expected value of the amount of matter of D that goes to x when this matter moves by J-jumps, and the number of J-jumps up to time t, Nt, follows a Poisson distribution with rate t:
u
(
x
,
t
)
=
∑
k
=
0
+
∞
F
(
k
)
(
x
,
D
)
e
-
t
t
k
k
!
.
It is well known that this function is the transition probability of a pseudo-Poisson process of intensity 1 (see [7, Chapter X]).
Moreover, from (1.7), we have that
ℍ
D
J
(
t
)
=
∑
k
=
0
+
∞
f
(
k
)
e
-
t
t
k
k
!
=
𝔼
(
f
(
N
t
)
)
is the expected value of the amount of matter of D that remains in D when this matter moves by J-jumps and the number of J-jumps up to time t follows a Poisson distribution with rate t.
2.3 The J-Heat Loss of D
Using (2.2), from (1.7) we can get the following expansion for the nonlocal J-heat loss of D inℝN at t (see [13]):
|
D
|
-
ℍ
D
J
(
t
)
=
∑
k
=
0
+
∞
P
(
J
∗
)
k
(
D
)
e
-
t
t
k
k
!
,
where we set P(J∗)0(D)=0.Using the above notation, we have
|
D
|
-
ℍ
D
J
(
t
)
=
𝔼
(
g
(
N
t
)
)
,
where g(k):=P(J∗)k(D).
2.4 The J-Heat Content and a Nonlocal Isoperimetric Inequality
The following isoperimetric inequality was given in [9]:
(2.14)
P
J
(
B
r
)
≤
P
J
(
D
)
for a ball
B
r
of radius
r
>
0
such that
|
B
r
|
=
|
D
|
.
The proof of this fact uses the Riesz rearrangement inequality, and as a consequence of [6, Theorem 1], in the case of a radially nonincreasing J having compact support Bδ(0), the ball of radius δ>0 centered at 0, the equality in (2.14) holds if and only if D is a ball of radius r with r>δ2.
From (2.14) we have the following result.
Corollary 2.4
Let J be radially nonincreasing and having compact support Bδ(0). For any bounded subset D⊂RN with |D|>δ2 we have
ℍ
D
J
(
t
)
≤
ℍ
B
r
J
(
t
)
for small
t
>
0
,
where Br is a ball such that |Br|=|D|.
Proof.
The result is true when D is a ball of radius r, so let us suppose that this is not the case. Then, on account of the comment after (2.4), we have
ℍ
B
r
J
(
0
)
=
|
D
|
=
ℍ
D
J
(
0
)
and
(
ℍ
B
r
J
)
′
(
0
)
=
-
P
J
(
B
r
)
>
-
P
J
(
D
)
=
(
ℍ
D
J
)
′
(
0
)
.
Then, by (1.9), we get
ℍ
D
J
(
t
)
<
ℍ
B
r
J
(
t
)
for small
t
>
0
,
and the result follows.∎
As a consequence of Corollary 2.4 and Proposition 2.1, we have the following characterization.
Corollary 2.5
Let J be radially nonincreasing and having compact support Bδ(0).The isoperimetric inequality (2.14) is equivalent to the inequality
∥
T
J
(
t
)
χ
D
∥
L
2
≤
∥
T
J
(
t
)
χ
B
r
∥
L
2
for small
t
>
0
,
with Br being a ball such that |Br|=|D| when r>δ2.
A similar result for the local case was proved in [11] (see also [4, 8]).
2.5 Convergence to the Heat Content when Rescaling the Kernel
For a subset D in ℝN with finite Lebesgue measure we will call
H
D
J
,
α
(
t
)
=
∫
D
u
(
x
,
t
)
the J-heat content of α-intensity of D,where u is the solution of
{
u
t
(
x
,
t
)
=
α
∫
ℝ
N
J
(
x
-
y
)
(
u
(
y
,
t
)
-
u
(
x
,
t
)
)
𝑑
y
,
(
x
,
t
)
∈
ℝ
N
×
[
0
,
∞
)
,
u
(
x
,
0
)
=
χ
D
(
x
)
,
x
∈
ℝ
N
.
Observe that HDJ(t) is the J-heat content of 1-intensity of D; also, HDJ,α(t)=HDJ(αt).
Let us consider the rescaled kernel for ε>0:
J
ε
(
x
)
=
1
ε
N
J
(
x
ε
)
,
and let
C
J
=
2
∫
ℝ
N
J
(
x
)
|
x
N
|
2
𝑑
x
.
Let vε be the solution of
{
(
v
ε
)
t
(
x
,
t
)
=
1
ε
2
[
J
ε
∗
v
ε
(
x
,
t
)
-
v
ε
(
x
,
t
)
]
,
x
∈
ℝ
N
,
t
∈
[
0
,
∞
)
,
v
ε
(
x
,
0
)
=
χ
D
(
x
)
,
x
∈
ℝ
N
.
By [3, Theorem 1.30], we have
lim
ε
→
0
+
∥
v
ε
-
v
∥
L
∞
(
ℝ
N
×
(
0
,
T
)
)
=
0
for every T>0, with v being the solution of the heat equation
{
v
t
(
x
,
t
)
=
1
C
J
Δ
v
(
x
,
t
)
,
x
∈
ℝ
N
,
t
∈
[
0
,
∞
)
,
v
(
x
,
0
)
=
χ
D
(
x
)
,
x
∈
ℝ
N
.
Set now u(x,t)=v(x,CJt). Then u verifies
{
u
t
(
x
,
t
)
=
Δ
u
(
x
,
t
)
,
x
∈
ℝ
N
,
t
∈
[
0
,
∞
)
,
u
(
x
,
0
)
=
χ
D
(
x
)
,
x
∈
ℝ
N
.
Hence for the solution uε of the problem
{
(
u
ε
)
t
(
x
,
t
)
=
C
J
ε
2
(
J
ε
∗
u
ε
(
x
,
t
)
-
u
ε
(
x
,
t
)
)
,
x
∈
ℝ
N
,
t
∈
[
0
,
∞
)
,
u
ε
(
x
,
0
)
=
χ
D
(
x
)
,
x
∈
ℝ
N
,
we have that
ℍ
D
J
ε
,
C
J
/
ε
2
(
t
)
=
∫
D
u
ε
(
x
,
t
)
𝑑
x
=
∫
D
v
ε
(
x
,
C
J
t
)
𝑑
x
and
lim
ε
→
0
∫
D
v
ε
(
x
,
C
J
t
)
𝑑
x
=
∫
D
v
(
x
,
C
J
t
)
𝑑
x
=
∫
D
u
(
x
,
t
)
𝑑
x
=
ℍ
D
(
t
)
.
Consequently, we have proved the following result (which is a restatement of Theorem 1.4 with the notation introduced in this section).
Theorem 2.6
For a subset D in RN with finite Lebesgue measure we have
lim
ε
→
0
+
ℍ
D
J
ε
,
C
J
/
ε
2
(
t
)
=
ℍ
D
(
t
)
for all
t
>
0
.
That is, if the jumps are rescaled to occur in a ball of radius ε and the intensity of the Poisson process that controls the intensity of the jumps is rescaled to the size CJ/ε2, then we are approaching, for ε small, the Gaussian heat content.
Remark 2.7
In [5] (see also [9]), it is shown that
(2.15)
lim
ε
↓
0
C
J
,
1
ε
P
J
ε
(
E
)
=
P
(
E
)
for a bounded set E⊂ℝN of finite perimeter,where
C
J
,
1
=
2
∫
ℝ
N
J
(
x
)
|
x
N
|
𝑑
x
.
Observe that
(
J
ε
*
)
k
=
[
(
J
*
)
k
]
ε
.
Then, by Theorem 1.2, for n≥1 we have
(
ℍ
D
J
ε
)
(
n
)
(
0
)
=
-
∑
k
=
1
n
(
n
k
)
(
-
1
)
n
-
k
P
[
(
J
*
)
k
]
ε
(
D
)
,
and consequently, by (2.15), there holds
lim
ε
→
0
1
ε
(
ℍ
D
J
ε
)
(
n
)
(
0
)
=
c
n
P
(
D
)
,
where
∑
k
=
1
n
(
n
k
)
c
k
=
1
2
∫
ℝ
N
(
J
*
)
n
(
x
)
|
x
N
|
d
x
,
that is,
c
n
=
-
1
2
∑
k
=
1
n
(
n
k
)
(
-
1
)
n
-
k
∫
ℝ
N
(
J
*
)
k
(
x
)
|
x
N
|
d
x
.
3 The Spectral Heat Content
Recall from Section 1 that in [3] the following nonlocal Dirichlet problem has been studied:
(3.1)
{
u
t
(
x
,
t
)
=
∫
ℝ
N
J
(
x
-
y
)
(
u
(
y
,
t
)
-
u
(
x
,
t
)
)
𝑑
y
,
(
x
,
t
)
∈
D
×
[
0
,
∞
)
,
u
(
x
,
t
)
=
0
,
(
x
,
t
)
∈
(
ℝ
N
∖
D
)
×
[
0
,
∞
)
,
u
(
x
,
0
)
=
u
0
(
x
)
,
x
∈
ℝ
N
,
and hence we defined
ℚ
D
J
(
t
)
:=
∫
D
u
(
x
,
t
)
𝑑
x
,
with u being the solution of (3.1) for the datum u0=χD.
Now, our task is to obtain the asymptotic expansion of ℚDJ(t).
First, we observe that
ℚ
D
J
(
0
)
=
|
D
|
.
For the second term in the expansion, it is easy to see that again
ℚ
D
J
(
t
)
+
(
ℚ
D
J
)
′
(
t
)
=
∫
D
∫
D
J
(
x
-
y
)
u
(
y
,
t
)
𝑑
y
𝑑
x
,
ℚ
D
J
(
0
)
+
(
ℚ
D
J
)
′
(
0
)
=
∫
D
∫
D
J
(
x
-
y
)
𝑑
y
𝑑
x
,
and hence
(
ℚ
D
J
)
′
(
0
)
=
-
P
J
(
D
)
.
Note that the first two terms in the expansions of ℚDJ(t) and of ℍDJ(t) coincide.Now the expression for thenext terms differs from that of the J-heat content. For example, now, instead of (2.8) we have
ℚ
D
J
(
0
)
+
2
(
ℚ
D
J
)
′
(
0
)
+
(
ℚ
D
J
)
′′
(
0
)
=
∫
D
∫
D
∫
D
J
(
x
-
y
)
J
(
y
-
z
)
𝑑
z
𝑑
y
𝑑
x
.
Hence,
(
ℚ
D
J
)
′′
(
0
)
=
∫
D
∫
D
∫
D
J
(
x
-
y
)
J
(
y
-
z
)
𝑑
z
𝑑
y
𝑑
x
+
2
P
J
(
D
)
-
|
D
|
,
which can be written as
(
ℚ
D
J
)
′′
(
0
)
=
∫
D
∫
D
∫
D
J
(
x
-
y
)
J
(
y
-
z
)
𝑑
z
𝑑
y
𝑑
x
+
∫
D
ℋ
∂
D
J
(
x
)
𝑑
x
,
is different from (2.9) in the term with three integrals.
Gathering this information, we now have
ℚ
D
J
(
t
)
=
|
D
|
-
P
J
(
D
)
t
+
1
2
∫
D
ℋ
∂
D
J
(
x
)
𝑑
x
t
2
+
1
2
∫
D
∫
D
∫
D
J
(
x
-
y
)
J
(
y
-
z
)
𝑑
z
𝑑
y
𝑑
x
t
2
+
O
(
t
3
)
as
t
↓
0
.
This proves Theorem 1.5.
,
Julio D. Rossi
