Abstract
In this article, we develop a new set of results based on a non-local gradient jointly inspired by the Riesz
1 Introduction
In the last decades, models based on differential equations are increasingly sharing their prominence with those based on integral or integro-differential equations. This is due to the fact that they can catch some information that the local ones cannot, such as long-range interactions or multi-scale behaviour; they usually require less regularity of the functions, allowing for more general admissible solutions. In fact, they can overcome some drawbacks of the local models since a typical feature is to be able to provide an effective modelling for discontinuities or singularities.
One such example is peridynamics, a non-local alternative model in solid mechanics proposed by Silling [68]; see also [47,69,70]. One of its goals was to unify elastic and singularity phenomena, such as fracture or cavitation. The development of this theory in the last years has been impressive. As general expositions, we can mention the review paper [43], the two books [36,48], and the collaborative handbooks [12,76]. Several aspects of these models have been studied such as localization [34,52,54], existence and regularity [35], computational issues [18,19], function spaces involved [50,53], or linear theories [23,27,29,51,65,75,79].
Accordingly, non-local models are gaining attention in the modelling of various phenomena in physics, biology, geometry, and more. As a consequence, it is required a more thorough mathematical analysis of the new objects and operators involved. Some of those objects are of diffusion type, where the fractional Laplacian stands out: this is an operator that generalizes the standard Laplacian to a degree of differentiability beyond derivatives of integer order (see, e.g. [1,14,15,22,56,60,62,63] among hundreds of possible references). Others, on the other hand, are of gradient type, which may provide a better scope for non-local vector calculus [7,16,20,21,24,52,54,66,67]. One of the features of both kinds of operators is that they require less regularity of the functions than their classical counterparts. In this article, we focus on the latter type, in particular, those called one-point gradients, or weighted non-local gradients in the terminology of [21]. These non-local gradients are usually written in terms of a kernel
A particular case of non-local gradient where this analysis has experienced a great interest since the works of Shieh and Spector [66,67] is Riesz’
where
This object has attracted a great interest in the last years, as will be reported below, mainly due to its better properties from a functional analysis perspective. However, the fact of being defined over the whole space is a remarkable drawback for applications in realistic physical models; furthermore, this feature obviously leads to some extra difficulties from a computational or numerical point of view.
Thus, given the interest of working with bounded domains, and inspired by (1) and (2), we propose the operator
as a particular case of (1), with
and
At this point, we ought to mention that referring
For example,
in (2), and
in (4). With this terminology, a natural classification of kernels arises, namely, those which are integrable (or locally integrable) and those which are not. Sometimes they are called kernels with a strong singularity or with a weak singularity (or no singularity at all), respectively. In our case, the operator
Another way of avoiding the instabilities while not necessarily considering singular kernels is provided by [46] through the analysis of non-radially symmetric kernels (in that case, defined over half-balls). This approach has been recently continued in [40] studying a Poincaré inequality, and in [25,42] for the traffic flow ahead of a vehicle. In [33], Poincaré inequalities are also studied under some assumptions on non-homogeneous kernels. Some other references for general heterogeneous kernels are [51,78], and in particular those with a variable horizon [26,71,74].
To put it into context, the fractional gradient (2), as recently addressed by several authors [7,8,16,44,61,64], seems to be the suitable notion, from a merely mathematical perspective, for such a differential object. In particular, it has been proved in [77] that formula (2) determines up to a multiplicative constant the unique object fulfilling some minimal consistency requirements from the physical and mathematical point of view, such as invariance under rotations and translations,
In the case of the non-local gradient
Regarding functional spaces, essential for calculus of variations, the one associated with Riesz fractional gradients is the Bessel space
There are, of course, many spaces between
and the norm
A great difference between
the fact that in
While [67] treated the existence of minimizers for convex scalar problems using the direct method of the calculus of variations, the following two papers make extensions and applications for vectorial problems: in [7], we showed that the concept of polyconvexity is also suitable in these problems, while in [44], it was shown the analogue for the concept of quasiconvexity. These notions, polyconvexity and quasiconvexity, are classical in the calculus of variations (see, e.g. [17]). In these three works, the functional to minimize is of the form
with the integrand
This article can be seen as a follow-up to other works in the search of a non-local model suitable for hyperelasticity but also introducing a theory applicable to other phenomena. Following previous works [9–11] by some of the authors of this article, we showed in [6] that (bond-based) peridynamics models based on energy functionals of the form
although defined for bounded domains, do not fit in non-linear solid mechanics, since very few local non-linear models are limit of non-local ones when
In this article, we propose a model that combines the good properties of the Riesz gradient and the space
To reproduce similar existence results for energy functionals like
it is necessary to study the functional space associated with the non-local gradient (4). To be precise, we define the space
where
We highlight as one of the main contributions of this work a non-local version of the fundamental theorem of calculus, which is obtained despite the lack of homogeneity and semigroup properties of the kernels involved. As in other frameworks, given that it allows us to recover a function from its non-local gradient, we see it as a versatile tool that may prove itself useful in other situations. In particular, it helps in overcoming the aforementioned problem of an absolute value affecting the inner integral in the seminorms we are considering, and therefore, it is a key ingredient in the process of obtaining several inequalities and embeddings.
Thus, this article can be regarded as a first step to explore properties in
Divergence: The classical divergence is the pointwise divergence. The fractional divergence is
while the non-local divergence is
Integration by parts: For
Fundamental theorem of calculus:
At this point, we mention the attempt to unify fractional and non-local theories recently explored in [21], in the context of a general vector calculus (following the earlier works [24,39]) and in [20], which focuses on non-local gradients. We also point out the work [31], where a different approach to the fractional fundamental theorem of calculus in dimension one is addressed, as well as a study of the function spaces involved.
The role of the Fourier transform in this analysis deserves a special mention. It was pointed out in [66] that the fractional gradient behaves nicely under Fourier transform:
The similarities with the fractional case show that a parallel theory could potentially be developed. Actually, it turns out that when restricted to
The outline of this article is the following. Section 2 fixes some notation used throughout the article. In Section 3, the new versions of non-local gradient, divergence, and integration by parts are established. We also define the associated function space
2 Notation
2.1 General notation
In all this work, we fix the dimension
The notation for Sobolev
We write
We denote by
We will use the multi-index notation: for
The vectors of the canonical basis of
The operation of convolution is denoted by
2.2 Fourier transform
The convention for the Fourier transform of a function
for
2.3 Radial functions
We recall the following definitions regarding radial functions.
Definition 2.1
We will say that
a function
a radial function
a function
It is known (see, e.g. [38, App. B.5]) that the Fourier transform of a radial (respectively, vector radial) function is radial (respectively, vector radial).
3 Function space: non-local gradient, divergence, and integration by parts
In this section, we define the non-local gradient and divergence, and state their basic properties, notably, the integration by parts. We also set the natural functional space associated with the non-local gradient. The framework is the following. As typical in non-local models [3–5,24,39,41], “boundary” conditions are usually of volumetric type. In our case, we fix a distance
The sets
Let
where the constant
and
There are constants
In fact, it will be apparent in the proof of Lemma 5.3 that condition (d) can be considerably weakened. Note, in addition, that
Given a function
is defined as follows:
whenever this limit exists.
The definitions of the non-local gradient and divergence for smooth functions are the following.
Definition 3.1
Set
Let
(9)Let
Notice that the integral in (9) is absolutely convergent because
and this last integral is absolutely convergent. Similarly,
Variants of these expressions have been used to study properties of non-local operators of this style [16,21,66,77]; notably, the Riesz fractional gradient
Note also that, for each
and similarly for the integral in (10), since
The operators of Definition 3.1 are dual operators in the sense of integration by parts. Actually, several versions of integration by parts formulas for related fractional or non-local operators have already appeared in the literature [16,24,54,77]. For the purposes of this work, we will use a particular case of [54, Th. 1.4], which, for convenience, we restate here in our context.
Proposition 3.1
Assume
The integration by parts formula of interest in this investigation is the following. Notice the presence of a boundary term, which is due to the fact that
Theorem 3.2
Suppose that
and these three integrals are absolutely convergent.
Proof
Denoting by
so
We have
with
By Proposition 3.1,
On the other hand,
Now, for each
and, since
and this last integral is absolutely convergent, as explained in (10).
Putting together the aforementioned formulas, we have obtained that
being the three integrals absolutely convergent. Finally, for each
since
Although we have proved
We now extend Definition 3.1(a) to a broader class of functions, namely,
Definition 3.2
Let
Let
The following result shows that the aforementioned definitions are independent of the sequence chosen.
Lemma 3.3
Let
Let
Proof
We prove (a), the proof of (b) being analogous. Let
By Theorem 3.2, for each
Now,
Assume that the limit
holds. Then, we would have proved that
Since the same reasoning can be done for
for all
It remains to justify limit (13). Due to the convergence
is in
Finally,
It is quite natural to consider the space of
Definition 3.3
We define the space
equipped with the norm
Notice that functions in
Similarly to Sobolev spaces, this space satisfies reflexivity and separability properties.
Proposition 3.4
Let
Proof
That
For the rest of the proof, we apply a standard argument; see, e.g. [52, Th. 2.1] for the non-local case and [13, Prop. 8.1] for the local case.
We have that the space
By Definitions 3.2 and 3.3, it is clear that
In the next result, we compare the spaces
Proposition 3.5
Let
Proof
The proof of (a) is obtained in a straightforward manner applying the known inclusions
Regarding (b), we first prove the corresponding inequality for smooth functions. Thus, let
where the last equality is obtained thanks to the odd symmetry of the integrand.
We recall that
for some constant
for some constant
Being the spaces
The inclusion of Proposition 3.5(b) is one of the main motivations for the definition of our space, since it roughly suggests that
4 Non-local fundamental theorem of calculus
We start by recalling the following classical representation theorem, which can be seen in [37, Lemma 7.14] or [59, Prop. 4.14].
Proposition 4.1
For every
This result may be understood as a fundamental theorem of calculus, in the sense that we recover a function from its gradient by integration; more precisely, by convolution. A fractional version of it, involving the Riesz fractional gradient, is also known [59,66]. This section is devoted to a novel non-local version of Proposition 4.1, where a function can be recovered from its non-local gradient
Our approach is inspired by the proofs of the fractional fundamental theorem of calculus previously referred in [59,66]. However, those partly rely on the semigroup properties of Riesz potentials, which our kernels do not enjoy. Therefore, our procedure is much more involved.
To begin with, we show that the kernel in the definition of
Definition 4.1
Define
as
In the following result,
Lemma 4.2
- (14)
Proof
We start with (a). The function
Since
so the existence of
We now show (b). We get immediately from (15) that
so (14) holds. As
In the following proposition, we write the non-local gradient as a convolution of the classical one with the kernel
Proposition 4.3
For every
and
Proof
Let
defined by
Let
and notice that
where
We estimate the integrand on the right-hand side. As
for some constant
which goes to 0 when
As a result, by using (17), we obtain that
provided that both limits exists, which is actually true as both integrals are absolutely convergent in
As this is true for every
and formula (16) is proved.
We have thus shown that
Proposition 4.3 shows that for
We continue by restating the main result of Section 5 (Theorem 5.9).
Proposition 4.4
There exists a function
Moreover,
We will study further properties of
Theorem 4.5
Let
Proof
Let
Next we make the changes of variables
By Proposition 4.4,
Thus, thanks to Proposition 4.1,
and the proof is complete.□
5 Existence of
V
δ
s
This section is devoted to the proof of Proposition 4.4, as well as to the derivation of further properties of
The idea of the proof is to convert equation (18) into
through Fourier transform (see Lemma B.1(c) for the Fourier transform of the right-hand side of (18)). Thus, the candidate for
In the first half of the section, we show that
As seen Section 1, it is illustrative to compare formula (19) with the fractional fundamental theorem of calculus in
Here,
where
We present another parallelism between
(recall (2)), we can write
where
is a (vectorial) Calderón-Zygmund operator of order zero. Thus, we have the decomposition
which splits
5.1 Positivity of
Q
ˆ
δ
s
and existence of
V
δ
s
as a distribution
We start with an analysis of the Fourier transform of the function
Lemma 5.1
The function
Proof
Given that
It remains to show that
where the superindex
while
and, hence,
This decay at infinity of
In the rest of the proof, we assume that
Consider a
Looking at the expression of
On the other hand, the function
is also faster than any negative power of
which allows us to conclude that
In the following result, we obtain relevant properties about
Proposition 5.2
Proof
The proof of part (a) comes directly from known facts in Fourier analysis. Indeed, as
Regarding (b), we have that
To show (c), we apply the Fourier transform to the expression
in the sense of distributions. Actually, by Young’s inequality for the convolution, we have that
Therefore, the integral defining
Then, we consider
As the function
equivalently,
Define now
and, in turn, equivalent to
in other words,
We recall from Lemma 4.2 that
Let
Therefore, as
Finally, we use that
As
The following calculation is useful for showing the positivity of
Lemma 5.3
For all
Proof
The integral is absolutely convergent since
and
Recall that
where
for each
We have that each term in the sum is positive; indeed, by splitting the integral in two through point
since
as
As can be seen from the aforementioned proof, the assumption that
Therefore, assumption (d) on
The following result shows the convergence of the truncations of
Lemma 5.4
The function
and we have the convergence
Proof
We recall from Lemma 4.2 that
Thus,
By using the notation
By the mean value theorem,
This shows that formula (26) defines a tempered distribution; moreover, by dominated convergence, we obtain that
as
We now show the positivity of
Proposition 5.5
Proof
Since
This is shown at the end of the proof. Assuming the validity of (27), by Lemma 5.3, we obtain
Now, the formula
holds in the sense of tempered distributions. Since both terms are actually functions, the equality holds as functions for almost every point. Moreover, since both functions are continuous, the equality holds everywhere. We then conclude that
Consequently, since
It remains to prove (27). By Lemma 4.2,
We have
so
We now compute
where we have used the odd symmetry. Now, by dominated convergence,
With this, we can conclude the existence of
Proposition 5.6
There exists a tempered distribution
Proof
Denote by
This function is in
As a result,
so the function in (29) is in
5.2 Existence of
V
δ
s
as a function
In this subsection, we prove that the distribution
We first need the following decay estimate for the derivatives of
Lemma 5.7
For every
Proof
In this proof, we use the letter
Express
By Leibniz’ formula,
Let
for some
We apply Faà di Bruno’s formula for the higher-order derivatives of a composition, and obtain that
where
We estimate the partial derivatives of
Now we show that
From Definition 4.1 and Lemma 4.2, we have that
for some
for some
which shows (33).
Now, by Leibniz’ formula, (32), and (33),
for some constant
for some constants
for some constants
From Proposition 5.2, we know that, for
so
Thus,
We conclude that, for
as desired.□
The decay estimate of Lemma 5.7 is not optimal. In fact, a more refined argument can possibly improve decay (33) and show that the bound
holds. With that estimate, an adaptation of the proof of Lemma 5.7 would yield
Nevertheless, the bound of Lemma 5.7 is enough for our purposes in Theorem 5.9. Before that, we need the following inverse Lipschitz estimate of the function
Lemma 5.8
For every
Proof
We divide the proof into four cases, according to the position of the points
Case 1:
we have
Case 2:
we have
Case 3:
We observe that the inverse of
where
for some constant
Now, by using (36),
Elementary geometry (see Figure 2) shows that
Assumption (35) asserts that one of the two options of (38) occurs, so
and, hence,
Finally, since
Going back to (37), we find that
Case 4:
Note first that inequality (40) cannot occur in dimension
Let
where
Assumption (40) implies that none of the cases of (38) occurs (hence none of the situations depicted in Figure 2), but the distance from the origin to the segment
For this particular
since, by construction of
and
Now we use the elementary inequality
to obtain that
so
By using (42) and (43), inequality (41) becomes
If we had assumed
so, in either case,
Now we use (39) and find that inequality (34) holds for
Position of the points
The curve
Finally, we present the main result of this section: its statement includes that of Proposition 4.4, shows that
Theorem 5.9
There exists a vector radial function
Furthermore, the following properties hold:
- (45)
There exists a Lipschitz bounded
(46)For any
For any
Proof
We first prove that there exists
We start as in the proof of [38, Prop. 2.4.8]. To see that
We take
On the one hand,
Write
Then
Having in mind (47), a classical result relating the decay of a function at infinity with the regularity of its Fourier transform (see, e.g. [38, Exercise 2.4.1]) shows that
Once we have shown that
so
Once we have that
Next, we show that the function
decays to zero at infinity faster than any negative power of
The terms
Looking at the expression of
On the other hand, the function
From expression (48), we can see that
We continue by proving that
We define
we obtain equality (46) (see Lemma B.1 for the inverse Fourier transform of the last term). That expression, together with the fact that
To show (a), we note that equality (45) is equivalent to the equality of its Fourier transforms. More precisely, the functions
which holds due to (44). The uniqueness of
Fact (b) has been proved when
which shows (c).
As for inequality (d), since
for a suitable constant
Part (b) of Theorem 5.9 shows that
6 Poincaré, Morrey, Trudinger, and Hardy inequalities
In this section, we will use the non-local fundamental theorem of calculus (Theorem 4.5) to prove inequalities in the spirit of Poincaré-Sobolev, Morrey, Trudinger, and Hardy. In those inequalities, we will need a boundary condition implying that the function vanishes in a tubular neighbourhood of
To describe more precisely that boundary condition, we recall the set
It is immediate to check that any
As in the space
The Poincaré-Sobolev inequality in
Theorem 6.1
Let
for every
Proof
By density, it is enough to prove the inequality for
Fix
where
Therefore, for every
A non-local Poincaré inequality is obtained as a corollary.
Theorem 6.2
Let
Proof
If
which is easily seen to exist. Indeed, if
which exists because
Once
Next we introduce a non-local analogue of Morrey’s inequality, whose fractional version was shown in [66, Th. 1.11]. Unlike their proof, which uses Morrey-type estimates of the Riesz transform, ours is based on the non-local fundamental theorem of calculus in this context (Theorem 4.5) together with the estimates of the kernel
Theorem 6.3
Let
and
In addition, any
Proof
The core of the proof consists in showing that
for all
Now define
For the first term, we have that by Hölder’s inequality,
since
Finally, so as to tackle the last term, by the fundamental theorem of calculus,
so
By Hölder’s inequality,
Since
since
Then, inequality (55) follows combining (57), (58), (59), and (60), as well as the inclusion
To show inequality (54) and the continuous inclusion
where
The limiting case
Theorem 6.4
Let
Proof
By a standard density argument, it is enough to prove the inequality for
By (51),
while by [37, Lemma 7.13], for some constants
Putting together these two inequalities, we obtain the conclusion.□
We mention that the constants
We end this section with the analogue of Hardy’s inequality. Its fractional version can be found in [66, Th. 1.9], whose proof (as well as the classical one [73]) is easily adapted to our context.
Theorem 6.5
Let
Proof
As before, it is enough to establish the inequality for
for some constants
7 Compact embeddings
In this section, we will use the non-local fundamental theorem of calculus (Theorem 4.5) to prove compact embeddings of the spaces
We start with the following Hölder estimate of the function
Lemma 7.1
There exists a constant
Proof
We first show that there exists a constant
On the one hand, we have that
On the other hand, for a fixed
Now, for
so
By integration, we obtain that
This yields (62).
To complete the proof, we take a rotation
A key ingredient of the desired compactness result is the application of the Fréchet-Kolmogorov criterion, for which the following estimate on the translations is crucial. The analogous result in the Sobolev case is classical [13, Prop. 9.3]. The next result is inspired by [16, Prop. 3.14], where they proved a fractional version when
Proposition 7.2
Let
(63)Let
Proof
By a standard density argument, it is enough to prove the result for
We start with case (a). Let us fix
and the proof is concluded in this case.
In the rest of the proof, we consider
By Theorem 5.9 (d), there exists
for all
where we have used Lemma 7.1. Next, we integrate and apply Fubini’s theorem to obtain
where we have applied Lemma 7.1 again. This completes the proof of (a).
Now we prove (b) by following the same lines as in case (a). As
By using Lemma 7.1 we find that, for some
which concludes the proof.□
The main result of this section is the following compact embedding, which is an analogue of the Rellich-Kondrachov theorem. See [67, Th. 2.2] for the fractional case; their proof uses Ascoli-Arzelà’s theorem to mollifiers of the sequence, while we prefer to invoke directly the Fréchet-Kolmogorov criterion. Recall the notation
Theorem 7.3
Let
for some
if
for every
if
Proof
Clearly,
The case
Let
We start with (a), so we assume
with
By using the interpolation inequality, (66), the triangular inequality and Theorem 6.1,
for some
As a result, the Fréchet-Kolmogorov criterion leads to the compactness of
Now we show (b), so we assume
for some
Of course, Theorem 6.3 yields additionally the compact inclusion of
8 Existence of minimizers and the Euler-Lagrange equation
In this final section, we prove the existence of minimizers of functionals of the form
under coercivity and convexity conditions. We also show the corresponding (non-local) Euler-Lagrange equations satisfied by the minimizers.
From now on,
Theorem 8.1
Let
For a.e.
There exist
for a.e.
Proof
Assumption (d) shows that the functional
Moreover,
A standard lower semicontinuity result for convex functionals (see, e.g. [32, Th. 7.5]) shows that
Therefore,
We finally show the Euler-Lagrange equation satisfied by any minimizer. The notation for partial derivatives is as follows:
Theorem 8.2
Let
There exist
for a.e.
If, in addition,
for a.e.
Proof
By using a standard argument, to show (68), it is enough to check that one can differentiate under the integral sign in the function
Now we make the assumption
This last integral is zero; indeed,
We combine this equality with (68) and apply the fundamental lemma of the calculus of variations to obtain that equality (69) holds for a.e.
Note that (69) imposes an a.e. equality in
Acknowledgements
We thank Davide Barbieri for useful discussions concerning the Fourier transform, and Hidde Schönberger for some comments on a previous version of the manuscript. We also thanks the referees for their comments.
-
Funding information: This work has been supported by the Agencia Estatal de Investigación of the Spanish Ministry of Research and Innovation, through projects PID2020-116207GB-I00 (J.C.B. and J.C.), PID2021-124195NB-C32 and the Severo Ochoa Programme for Centres of Excellence in R&D CEX2019-000904-S (C.M.-C.), by Junta de Comunidades de Castilla-La Mancha through project SBPLY/19/180501/000110 and European Regional Development Fund 2018/11744 (J.C.B. and J.C.), by the Madrid Government (Comunidad de Madrid, Spain) under the multi-annual Agreement with UAM in the line for the Excellence of the University Research Staff in the context of the V PRICIT (Regional Programme of Research and Technological Innovation) (C.M.-C.), by the ERC Advanced Grant 834728 (C.M.-C.), and by Fundación Ramón Areces (J.C.).
-
Conflict of interest: The authors declare no conflict of interest.
Appendix A
V
δ
s
when
n
=
1
In this appendix, we explain the necessary changes in the proof of Theorem 5.9(b) when
Lemma A.1
Let
(a) The function
(A1)and we have the convergence
(A2)The function
(A3)can be identified with the tempered distribution
we have the convergence
(A4)and
(A5)
Proof
We start with (a). Let us see that formula (A1) defines a tempered distribution. By Propositions 5.2 and 5.5, there exists
Thus, by the mean value theorem,
On the other hand, in the proof of Theorem 5.9(b), we saw that
Consequently,
Estimates (A6) and (A7) show that
As before, the fact that
As we saw in (A6)–(A7), the function
which justifies the identification of the function
Thus, (a) is proved. The proof of (b) is analogous, and we only write the details for the expression of
because of odd symmetry. As the function
We are in a position to prove Theorem 5.9(b).
Proof of Theorem 5.9(b) when n = 1
It remains to show that
Since
and, on the other hand,
which is integrable in
It remains to show that
This latter function is known to be bounded. Therefore,
B Fourier analysis results
In this appendix, we collect several Fourier analysis results needed throughout the article. They are possibly known to experts, but we have not found a precise reference.
First, we compute the Fourier transform of the vectorial version of the Riesz potential.
Lemma B.1
If
If
Proof
Fix
Thus,
On the other hand, by standard properties of the Fourier transform, and, in particular, by (21),
Putting together (A8) and (A9) we obtain the conclusion of (a).
Now we present the proof of (b). We recall formula (20) of the fractional version of the fundamental theorem of calculus: for every
Next, we take Fourier transform and use the formula for the convolution of a distribution with a Schwartz function:
where we have used the explicit expression for Fourier transform of
Since that equality holds for every
Finally, we prove (c). Since
Since this is true for every
Therefore, there exists a function
On the other hand,
Now, we recall the following definitions and properties about the convolution and Fourier transform of tempered distributions. Recall from Section 2.2 the notation
Remark B.1
Let
Assume that
We have that
Finally, we show that the product property of the Fourier transform of a convolution also holds for tempered distributions.
Lemma B.2
Let
Proof
Firstly, we recall that the convolution
Now, by definition of the Fourier transform in the sense of distributions, for every
Next, by the Fourier transform of a convolution (of a distribution with a Schwartz function),
since
As
Consequently, the desired formula holds.□
References
[1] N. Abatangelo and E. Valdinoci, Getting acquainted with the fractional Laplacian, in Contemporary research in elliptic PDEs and related topics, vol. 33 Springer INdAM Series, Springer, Cham, 2019, pp. 1–105. 10.1007/978-3-030-18921-1_1Search in Google Scholar
[2] R. A. Adams, Sobolev spaces, Pure and Applied Mathematics, vol. 65, Academic Press, New York-London, 1975. Search in Google Scholar
[3] B. Aksoylu and T. Mengesha, Results on non-local boundary value problems, Numer. Funct. Anal. Optim. 31 (2010), 1301–1317. 10.1080/01630563.2010.519136Search in Google Scholar
[4] B. Aksoylu and M. L. Parks, Variational theory and domain decomposition for non-local problems, Appl. Math. Comput. 217 (2011), 6498–6515. 10.1016/j.amc.2011.01.027Search in Google Scholar
[5] F. Andreu, J. M. Mazón, J. D. Rossi, and J. Toledo, A non-local p-Laplacian evolution equation with non-homogeneous Dirichlet boundary conditions, SIAM J. Math. Anal. 40 (2009), 1815–1851. 10.1137/080720991Search in Google Scholar
[6] J. C. Bellido, J. Cueto, and C. Mora-Corral, Bond-based peridynamics does not converge to hyperelasticity as the horizon goes to zero, J. Elasticity 141 (2020), 273–289. 10.1007/s10659-020-09782-9Search in Google Scholar
[7] J. C. Bellido, J. Cueto, and C. Mora-Corral, Fractional Piola identity and polyconvexity in fractional spaces, Ann. I. H. Poincaré - AN 37 (2020), 955–981. 10.1016/j.anihpc.2020.02.006Search in Google Scholar
[8] J. C. Bellido, J. Cueto, and C. Mora-Corral, Γ-convergence of polyconvex functionals involving s-fractional gradients to their local counterparts, Calc. Var. Partial Differential Equations 60 (2021), Paper no. 7, 29. 10.1007/s00526-020-01868-5Search in Google Scholar
[9] J. C. Bellido and C. Mora-Corral, Existence for non-local variational problems in peridynamics, SIAM J. Math. Anal. 46 (2014), 890–916. 10.1137/130911548Search in Google Scholar
[10] J. C. Bellido and C. Mora-Corral, Lower semicontinuity and relaxation via Young measures for non-local variational problems and applications to peridynamics, SIAM J. Math. Anal. 50 (2018), 779–809. 10.1137/17M1114181Search in Google Scholar
[11] J. C. Bellido, C. Mora-Corral, and P. Pedregal, Hyperelasticity as a Γ-limit of peridynamics when the horizon goes to zero, Calc. Var. Partial Differential Equations 54 (2015), 1643–1670. 10.1007/s00526-015-0839-9Search in Google Scholar
[12] F. Bobaru, J. T. Foster, P. H. Geubelle, and S. A. Silling, eds., Handbook of peridynamic modeling, Advances in Applied Mathematics, CRC Press, Boca Raton, FL, 2017. 10.1201/9781315373331Search in Google Scholar
[13] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. 10.1007/978-0-387-70914-7Search in Google Scholar
[14] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007), 1245–1260. 10.1080/03605300600987306Search in Google Scholar
[15] W. Chen, Y. Li, and P. Ma, The fractional Laplacian, World Scientific Publishing Co. Pte. Ltd. Hackensack, NJ, 2020. Search in Google Scholar
[16] G. E. Comi and G. Stefani, A distributional approach to fractional Sobolev spaces and fractional variation: existence of blow-up, J. Funct. Anal. 277 (2019), 3373–3435. 10.1016/j.jfa.2019.03.011Search in Google Scholar
[17] B. Dacorogna, Direct methods in the calculus of variations, Applied Mathematical Sciences, 2nd ed., vol. 78, Springer, New York, 2008. Search in Google Scholar
[18] M. D’Elia, Q. Du, C. Glusa, M. Gunzburger, X. Tian and Z. Zhou, Numerical methods for non-local and fractional models, Acta Numer. 29 (2020), 1–124. 10.1017/S096249292000001XSearch in Google Scholar
[19] M. D’Elia, Q. Du and M. D. Gunzburger, Recent progress in mathematical and computational aspects of peridynamics, in: Handbook of Nonlocal Continuum Mechanics for Materials and Structures, G. Z. Voyiadjis, ed., Springer, 2019. 10.1007/978-3-319-58729-5_30Search in Google Scholar
[20] M. D’Elia, M. Gulian, T. Mengesha, and J. M. Scott, Connections between non-local operators: From vector calculus identities to a fractional Helmholtz decomposition, Fract. Calc. Appl. Anal. 25 (2022), 2488–2531. 10.1007/s13540-022-00038-ySearch in Google Scholar
[21] M. D’Elia, M. Gulian, H. Olson, and G. E. Karniadakis, Towards a unified theory of fractional and non-local vector calculus, Fract. Calc. Appl. Anal. 24 (2021), 1301–1355. 10.1515/fca-2021-0057Search in Google Scholar
[22] M. D’Elia and M. Gunzburger, The fractional Laplacian operator on bounded domains as a special case of the non-local diffusion operator, Comput. Math. Appl. 66 (2013), 1245–1260. 10.1016/j.camwa.2013.07.022Search in Google Scholar
[23] Q. Du, M. Gunzburger R. B. Lehoucq, and K. Zhou, Analysis of the volume-constrained peridynamic Navier equation of linear elasticity, J. Elasticity 113 (2013), 193–217. 10.1007/s10659-012-9418-xSearch in Google Scholar
[24] Q. Du, M. Gunzburger, R. B. Lehoucq, and K. Zhou, A non-local vector calculus, non-local volume-constrained problems, and non-local balance laws, Math. Models Methods Appl. Sci. 23 (2013), 493–540. 10.1142/S0218202512500546Search in Google Scholar
[25] Q. Du, K. Huang J. Scott, and W. Shen, A Space-time Nonlocal Traffic Flow Model: Relaxation Representation and Local Limit, 2022. ArXiv:http://arXiv.org/abs/2211.00796. Search in Google Scholar
[26] Q. Du, T. Mengesha, and X. Tian, Fractional Hardy-type and trace theorems for non-local function spaces with heterogeneous localization, Anal. Appl. (Singap.) 20 (2022), 579–614. 10.1142/S0219530521500329Search in Google Scholar
[27] Q. Du and X. Tian, Stability of non-local Dirichlet integrals and implications for peridynamic correspondence material modeling, SIAM J. Appl. Math. 78 (2018), 1536–1552. 10.1137/17M1139874Search in Google Scholar
[28] J. Duoandikoetxea, Fourier analysis, vol. 29 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2001. Search in Google Scholar
[29] E. Emmrich and O. Weckner, On the well-posedness of the linear peridynamic model and its convergence towards the Navier equation of linear elasticity, Commun. Math. Sci. 5 (2007), 851–864. 10.4310/CMS.2007.v5.n4.a6Search in Google Scholar
[30] A. Evgrafov and J. C. Bellido, From non-local Eringen’s model to fractional elasticity, Math. Mech. Solids, 24 (2019), 1935–1953. 10.1177/1081286518810745Search in Google Scholar
[31] X. Feng and M. Sutton, A new theory of fractional differential calculus, Analysis and Applications 19 (2021), 715–750. 10.1142/S0219530521500019Search in Google Scholar
[32] I. Fonseca and G. Leoni, Modern Methods in the Calculus of Variations: Lp Spaces, Springer, New York, 2007. Search in Google Scholar
[33] M. Foss, Nonlocal Poincaré inequalities for integral operators with integrable non-homogeneous kernels, 2019, ArXiv preprint 1911.10292.Search in Google Scholar
[34] M. Foss, P. Radu, and Y. Yu, Convergence analysis and numerical studies for linearly elastic peridynamics with Dirichlet-type boundary conditions, J. Peridyn. Nonlocal Model. 5 (2023) 275–310.10.1007/s42102-021-00074-xSearch in Google Scholar
[35] M. D. Foss, P. Radu, and C. Wright, Existence and regularity of minimizers for non-local energy functionals, Differential Integral Equations 31 (2018), 807–832. 10.57262/die/1537840870Search in Google Scholar
[36] W. H. Gerstle, Introduction to practical peridynamics, Frontier Research in Computation and Mechanics of Materials and Biology, vol. 1, World Scientific Publishing Co Pte. Ltd. Hackensack, NJ, 2016. 10.1142/9687Search in Google Scholar
[37] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. 10.1007/978-3-642-61798-0Search in Google Scholar
[38] L. Grafakos, Classical Fourier analysis, Graduate Texts in Mathematics, 2nd edn. , vol. 249, Springer, New York, 2008. 10.1007/978-0-387-09432-8Search in Google Scholar
[39] M. Gunzburger and R. B. Lehoucq, A non-local vector calculus with application to non-local boundary value problems, Multiscale Model. Simul. 8 (2010), 1581–1598. 10.1137/090766607Search in Google Scholar
[40] Z. Han and X. Tian, Nonlocal Half-ball Vector Operators on Bounded Domains: Poincare Inequality and Its Applications, 2022, ArXiv:http://arXiv.org/abs/2212.13720. Search in Google Scholar
[41] B. Hinds and P. Radu, Dirichlet’s principle and wellposedness of solutions for a non-local p-Laplacian system, Appl. Math. Comput. 219 (2012), 1411–1419. 10.1016/j.amc.2012.07.045Search in Google Scholar
[42] K. Huang and Q. Du, Asymptotically Compatibility of a Class of Numerical Schemes for a Nonlocal Traffic Flow Model, 2023, ArXiv:http://arXiv.org/abs/2301.00803. Search in Google Scholar
[43] A. Javili, R. Morasata, E. Oterkus, and S. Oterkus, Peridynamics Review, Math. Mech. Solids 24 (2019), 3714–3739. 10.1177/1081286518803411Search in Google Scholar
[44] C. Kreisbeck and H. Schönberger, Quasiconvexity in the fractional calculus of variations: characterization of lower semicontinuity and relaxation, Nonlinear Anal. 215 (2022), 112625, 26. 10.1016/j.na.2021.112625Search in Google Scholar
[45] S. Lang, Real Analysis, 2nd edn., Addison-Wesley, Reading MA, 1983. Search in Google Scholar
[46] H. Lee and Q. Du, Nonlocal gradient operators with a nonspherical interaction neighborhood and their applications, ESAIM Math. Model Numer. Anal. 54 (2020), 105–128. 10.1051/m2an/2019053Search in Google Scholar
[47] R. B. Lehoucq and S. A. Silling, Force flux and the peridynamic stress tensor, J. Mech. Phys. Solids 56 (2008), 1566–1577. 10.1016/j.jmps.2007.08.004Search in Google Scholar
[48] E. Madenci and E. Oterkus, Peridynamic Theory and Its Applications, Springer, New York, NY, 2014. 10.1007/978-1-4614-8465-3Search in Google Scholar
[49] C. Martínez, M. Sanz, and F. Periago, Distributional fractional powers of the Laplacean. Riesz potentials, Studia Math. 135 (1999), 253–271. Search in Google Scholar
[50] T. Mengesha, Nonlocal Korn-type characterization of Sobolev vector fields, Commun. Contemp. Math. 14 (2012), 1250028, 28. 10.1142/S0219199712500289Search in Google Scholar
[51] T. Mengesha and Q. Du, Nonlocal constrained value problems for a linear peridynamic Navier equation, J. Elasticity 116 (2014), 27–51. 10.1007/s10659-013-9456-zSearch in Google Scholar
[52] T. Mengesha and Q. Du, On the variational limit of a class of non-local functionals related to peridynamics, Nonlinearity 28 (2015), 3999–4035. 10.1088/0951-7715/28/11/3999Search in Google Scholar
[53] T. Mengesha and Q. Du Characterization of function spaces of vector fields and an application in non-linear peridynamics, Nonlinear Anal. 140 (2016), 82–111. 10.1016/j.na.2016.02.024Search in Google Scholar
[54] T. Mengesha and D. Spector, Localization of non-local gradients in various topologies, Calc. Var. Partial Differential Equations 52 (2015), 253–279. 10.1007/s00526-014-0711-3Search in Google Scholar
[55] Y. Mizuta, Potential Theory in Euclidean Spaces, GAKUTO International Series, vol. 6, Mathematical Sciences and Applications, Gakkōtosho Co., Ltd, Tokyo, 1996. Search in Google Scholar
[56] G. Molica Bisci, V. D. Radulescu and R. Servadei, Variational methods for non-local fractional problems, vol. 162 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2016. 10.1017/CBO9781316282397Search in Google Scholar
[57] A. C. Ponce, An estimate in the spirit of Poincaré’s inequality, J. Eur. Math. Soc. 6 (2004), 1–15. 10.4171/JEMS/1Search in Google Scholar
[58] A. C. Ponce, A new approach to Sobolev spaces and connections to Γ-convergence, Calc. Var. Partial Differential Equations 19 (2004), 229–255. 10.1007/s00526-003-0195-zSearch in Google Scholar
[59] A. C. Ponce, Elliptic PDEs, measures and capacities. From the Poisson equations to non-linear Thomas-Fermi problems, vol. 23 of EMS Tracts in Mathematics European Mathematical Society (EMS), Zürich, 2016. 10.4171/140Search in Google Scholar
[60] C. Pozrikidis, The Fractional Laplacian, CRC Press, Boca Raton, FL, 2016. 10.1201/b19666Search in Google Scholar
[61] J. F. Rodrigues and L. Santos, On non-local variational and quasi-variational inequalities with fractional gradient, Appl. Math. Optim. 80 (2019), 835–852. 10.1007/s00245-019-09610-0Search in Google Scholar
[62] X. Ros-Oton, Nonlocal elliptic equations in bounded domains: A survey, Publ. Mat. 60 (2016), 3–26. 10.5565/PUBLMAT_60116_01Search in Google Scholar
[63] X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl. 101 (2014), 275–302. 10.1016/j.matpur.2013.06.003Search in Google Scholar
[64] A. Schikorra, D. Spector, and J. Van Schaftingen, An L1-type estimate for Riesz potentials, Rev. Mat. Iberoam. 33 (2017), 291–303. 10.4171/RMI/937Search in Google Scholar
[65] J. Scott and T. Mengesha, A fractional Korn-type inequality, Discrete Contin. Dyn. Syst. 39 (2019), 3315–3343. 10.3934/dcds.2019137Search in Google Scholar
[66] T.-T. Shieh and D. E. Spector, On a new class of fractional partial differential equations, Adv. Calc. Var. 8 (2015), 321–336. 10.1515/acv-2014-0009Search in Google Scholar
[67] T.-T. Shieh and D. E. Spector, On a new class of fractional partial differential equations II, Adv. Calc. Var. 11 (2018), 289–307. 10.1515/acv-2016-0056Search in Google Scholar
[68] S. A. Silling, Reformulation of elasticity theory for discontinuities and long-range forces, J. Mech. Phys. Solids 48 (2000), 175–209. 10.1016/S0022-5096(99)00029-0Search in Google Scholar
[69] S. A. Silling, M. Epton, O. Weckner, J. Xu, and E. Askari, Peridynamic states and constitutive modeling, J. Elasticity 88 (2007), 151–184. 10.1007/s10659-007-9125-1Search in Google Scholar
[70] S. A. Silling and R. B. Lehoucq, Peridynamic theory of solid mechanics, in: Advances in Applied Mechanics, H. Aref and E. van der Giessen, Eds., vol. 44 Elsevier, Amsterdam, 2010, pp. 73–168. 10.1016/S0065-2156(10)44002-8Search in Google Scholar
[71] S. A. Silling, D. J. Littlewood, and P. Seleson, Variable horizon in a peridynamic medium, J. Mech. Mater. Struct. 10 (2015), 591–612. 10.2140/jomms.2015.10.591Search in Google Scholar
[72] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. Search in Google Scholar
[73] E. M. Stein and G. Weiss, Fractional integrals on n-dimensional Euclidean space, J. Math. Mech. 7 (1958), 503–514. 10.1512/iumj.1958.7.57030Search in Google Scholar
[74] Y. Tao, X. Tian, and Q. Du, Nonlocal models with heterogeneous localization and their application to seamless local-non-local coupling, Multiscale Model. Simul. 17 (2019), 1052–1075. 10.1137/18M1184576Search in Google Scholar
[75] X. Tian and Q. Du, Analysis and comparison of different approximations to non-local diffusion and linear peridynamic equations, SIAM J. Numer. Anal. 51 (2013), 3458–3482. 10.1137/13091631XSearch in Google Scholar
[76] G. Z. Voyiadjis, ed., Handbook of Nonlocal Continuum Mechanics for Materials and Structures, Springer, Cham, 2019. 10.1007/978-3-319-58729-5Search in Google Scholar
[77] M. Šilhavý, Fractional vector analysis based on invariance requirements (critique of coordinate approaches), Contin. Mech. Thermodyn. 32 (2020), 207–228. 10.1007/s00161-019-00797-9Search in Google Scholar
[78] X. Yu, Y. Xu, and Q. Du, Numerical simulation of singularity propagation modeled by linear convection equations with spatially heterogeneous non-local interactions, J. Sci. Comput, 92 (2022), Paper No. 59, 24. 10.1007/s10915-022-01915-7Search in Google Scholar
[79] K. Zhou and Q. Du Mathematical and numerical analysis of linear peridynamic models with non-local boundary conditions, SIAM J. Numer. Anal. 48 (2010), 1759–1780. 10.1137/090781267Search in Google Scholar
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