Abstract
We introduce three representation formulas for the fractional p-Laplace operator in the whole range of parameters 0 < s < 1 and 1 < p < ∞. Note that for p ≠ 2 this a nonlinear operator. The first representation is based on a splitting procedure that combines a renormalized nonlinearity with the linear heat semigroup. The second adapts the nonlinearity to the Caffarelli-Silvestre linear extension technique. The third one is the corresponding nonlinear version of the Balakrishnan formula. We also discuss the correct choice of the constant of the fractional p-Laplace operator in order to have continuous dependence as p → 2 and s → 0+, 1−.
A number of consequences and proposals are derived. Thus, we propose a natural spectral-type operator in domains, different from the standard restriction of the fractional p-Laplace operator acting on the whole space. We also propose numerical schemes, a new definition of the fractional p-Laplacian on manifolds, as well as alternative characterizations of the Ws, p(ℝn) seminorms.
1 Introduction
In recent years, nonlocal differential operators and equations have attracted an increasing amount of attention. In particular, the fractional Laplacian is defined for suitable functions u(x), x ∈ ℝn, as
where s ∈ (0, 1), cn, s > 0 is a normalization constant and PV denotes that the integral is taken in the principal value sense, cf. the classical references [39, 51]. It appears (but for the constant factor) as the subdifferential of the fractional energy
that is related to the definition of the standard norm of the space Ws,2(ℝn). Besides, there are several known representation formulas that have proven to be quite convenient in tackling problems involving fractional operators like (1.1), see the list in [38] where a rather thorough discussion is made. Three of those representations are quite relevant and will be re-examined here in a nonlinear context. They are the so called semigroup (or Bochner’s subordination) formula; the Caffarelli-Silvestre extension problem, that allows to write the nonlocal operator as in terms of a local problem in one extra dimension; and Balakrishnan’s formula.
(i) A first representation was given by Bochner (see [10]). In modern form, it says that for a smooth enough function u the following formula holds
where etΔ denotes the heat semigroup, i. e., we take etΔ[u](x) ≔ w(x, t), where w is the solution of the problem
(ii) On the other hand, the fractional Laplacian can be written in local terms since Caffarelli and Silvestre showed in [19] that the extension problem
allows to compute the fractional Laplacian (−Δ)s u as a weighted Neumann limit of v
(iii) A further representation for the fractional Laplacian was given by Balakrishnan whose formula reads (see [3])
The three representations have proved to be very useful both in theory and applications, and they have given rise to a wide literature.
Aim and main results
The aim of this paper is to present analogous representations in the nonlinear context of the fractional p-Laplacian, and we comment on their usefulness in different applications. We first recall that the fractional p-Laplacian is usually defined as follows
where p ∈ (1, ∞) and s ∈ (0, 1), we write Φp(t)=∣t∣p−2t, and C1=C1(n, s, p) > 0 is a constant. The fractional p-Laplacian is the subdifferential of a convex functional called the (s, p)-energy (or Gagliardo seminorm)
defined for those functions in L2(ℝn) with finite (s, p)-energy, see e.g. [47, 55]. We also recall that this operator has a dense domain in L2(ℝn) containing smooth enough functions. Actually,
Our main result is the introduction of three equivalent representation formulas for
where C2 > 0. The precise study of this nonlinear representation occupies Section 2, see Theorem 2.1.
Second, we consider an extension operator Es, p related to a local problem in ℝn × (0, ∞), and such that the following representation holds:
where C3 > 0. The precise study of this nonlinear representation occupies Section 3, see Theorem 3.1.
Lastly, the fractional p-Laplacian is also equivalently given by the following nonlinear Balakrishnan’s formula
where C4 > 0. This will be studied in Section 4, see Theorem 4.1.
The choice of the constant C1 is indifferent in most cases and can be fixed as 1, but a precise value C1(n, s, p) can be important in writing some exact formulas or in the limit cases s → 0+, s → 1−, or p → 2. In Section 5 constants we will discuss the choice of this constant and also give an explicit relation with C2, C3, and C4, which turn out to be dimensionally free after a suitable choice of C1.
The three above representations succeed in replicating the linear formulas in the nonlinear setting, which was unexpected. One may wonder about the complexity introduced by the nonlinear term Φp(u(x) − u(y)). In that sense, we have the following comments. Firstly, we note that the operator used in each case (etΔ, Es, p and Δ(t−Δ)−1) is independent of the point x where
Representation in bounded domains. As a first consequence of the above results, we remark that the first representation of the fractional p-Laplacian naturally suggests a version of the operator defined in bounded domains Ω of ℝn. Given u:Ω → ℝ we define a new operator
where
This operator is defined for any p > 1 and generalises the Spectral Fractional Laplacian, which is the well-known case p = 2. In Section 6 new spectral we will show this operator is well-defined and different from the restriction of
Further extensions and comments. Section 7 addresses the topic of numerical analysis for the fractional p-Laplacian starting from our representation formulas. Section 8 is devoted to comment on some extensions and applications of our results. In Section 9 commentandproblems, we present some comments and open problems that we find interesting as well as a brief intuition about our results. Finally, we include an appendix with technical results.
Comments on related literature. Some references for stationary boundary value problems involving the fractional p-Laplace operator that we study here: Ishii and Nakamura [35] treat the first boundary value problem; Lindgren and Lindqvist [41, 42] establish the fractional eigenvalues, Perron’s Method and Wiener’s theorem. Regularity was studied in [14,15,16,34,37,40] among others. The equivalence of weak and viscosity solutions in boundary value problems involving the fractional p-Laplacian has been recently treated in [36] in the homogeneous case and [5] in the non-homogeneous case. The parabolic problem has been studied in bounded domains: Mazón, Rossi, and Toledo [47], Puhst [50], and Vázquez [57]; and also in the whole space ℝn by M. Strömqvist [53] and Vázquez [55].
We will make specific comments on the semigroup formula, the extension problem and the Balakrishnan’s formula in the corresponding sections. Other applications, in particular numerical studies are mentioned in Section 7 and Section 8.
The literature contains works on other types of operators that are also known as “nonlocal or fractional p-Laplacian operators”. We are not treating such versions in our paper since they lead to quite different objects, tools and results. For instance, in the book by Andreu-Vaillo, Mazón, Rossi, Toledo, [1], the integro-differential operator is defined on the basis of integrable kernels. Cipriani and Grillo proposed in [22] another type of nonlinear and nonlocal operator defined via the subordination formula applied directly to a nonlinear semigroup. We show in the last section that this method leads to an operator that is different from
As a final comment, we recall that the standard p-Laplace operator (s = 1) has been widely studied both in the stationary and evolution equations. Let us mention some classical monographs by DiBenedetto [31], Lindqvist [43], that contain further references. The monograph [56, Section 11] contains a summary of the evolution results.
2 Semigroup representation
We present an interesting representation of the operator using the heat semigroup. Let us use the notation etΔ for this semigroup, i.e., given f : ℝn → ℝ we denote by w(x, t) = etΔ[f](x) the solution of
The following is our first main result.
Theorem 2.1
Let s ∈ (0, 1) and p ∈ (1, +∞). Assume
where
Note that, when p = 2, we have that
so that, a suitable choice of C1(n, s, p) allows to recover Bochner’s formula (1.3) from (2.1). Notice that Γ(−s) < 0 while C2 > 0.
Our proof is based on the scaling properties of the representation formula
We use here it in combination with the nonlinearity, plus a desingularization step that is done by subtraction inside the argument of the nonlinearity. In this way we obtain the splitting representation (2.1) of the fractional p-Laplacian.
Proof of Theorem 2.1
Let ε > 0 and consider the bounded function gε : ℝn × ℝn → 0 given by gε(x, y) = Φp(u(x) − u(y)) χ∣x−y∣ > ε. Fix x ∈ ℝn and consider the heat semigroup associated to gε(x, ·), i.e.
Taking z=x we have
Let now δ > 0 and integrate the quantity above with respect the singular measure
where we have used Fubini’s theorem to change the order of integration. We use now the change of variables t = ∣x − y∣2/(4τ) to get
Note that, since gε(x, y)=0 if ∣x − y∣ < ε, we can consider the above expression to be integrated in ℝn∖ Bε(x). Now, we want to let δ → 0+ to get
Step 1: Let either p ∈ [2, ∞) or p ∈ (1, 2) and ∇ u(x)≠ 0.
Passing to the limit in the right hand side is trivial since the integrand converges pointwise and monotonically to
where we have used either Lemma A.1 or Lemma A.2 with
and the change of variables
where the bound comes from the fact that (1 − s)p/2 > 0 since s ∈ (0, 1).
Now we want to take limits as ε → 0. In the right hand side, we use either Lemma A.1 or Lemma A.2 with K(z) = ∣z∣−n−spχ∣z∣<ε(z) to get
While in the left hand side we use either Lemma A.1 or Lemma A.2 with
and for t = 1,
Then
Clearly, the second term in the last estimate goes to zero as ε → 0. For the first term, let
For the first term, clearly fε → 0 a.e in t ∈ [0, ∞),
and
so by the Dominated Convergence Theorem this term also converges to zero as ε → 0.
Step 2: Now let
so that
which is finite since
3 The new extension problem
The aim of this section is to write corresponding formulas in the present nonlinear setting following the ideas introduced in the linear setting by Caffarelli and Silvestre [19].
For s ∈ (0, 1) and p ∈ (1, +∞) consider the extension problem
which extends the usual theory where p = 2. We will show that a classical solution of this problem can be recovered by convolution with an explicit Poisson kernel
Finally, let us define the extension operator Es, p [f] ≔ U through formula (3.2). With this extension operator we can give a representation formula for the (s, p)-Laplacian.
Theorem 3.1
Let s ∈ (0, 1) and p ∈ (1, +∞). Assume
where the constant is given by
Actually, for p = 2 we have that
recovering the classical result of [19] for the fractional Laplacian.
Remark 3.1
Notice that by L’Hôpital’s rule,
In order to prove Theorem 3.1 extension we need an important lemma that gives an alternative representation of the extension operator based on the one by Stinga and Torrea ([52]) for fractional powers of general linear operators.
Lemma 3.1
Let s ∈ (0, 1) and p ∈ (1, +∞). If f ∈ Cb(ℝn) then Es, p[f] is a classical solution of (3.1). Furthermore, it can be alternatively represented by
Proof
We proceed in several steps.
We will check here that the pointwise formulas (3.4) and (3.2) are well defined for y > 0.
First, for (3.4), we use the explicit representation of etΔ, and the fact that f is bounded
The last term is clearly finite since sp/2 > 0 and
Second, for, (3.2)
which again is finite since
Now we check that both pointwise formulas are equivalent. Notice that, by Step 1, all the integrals below are well defined, and they allow us to do Fubini to change the order of integration. Then,
Step 3. Now we check that formulas (3.4) and (3.2) satisfy (3.1). We start with the boundary condition. To do so, it is sufficient to show that
For x≠=0 it is easy to check from (3.2) that we have
It is also standard to get
Finally, it is easy to compute (see for example [19]) that for all y > 0 we have
We have that U is a pointwise solution of the extension equation.
Proof of Theorem 3.1
We need to check (3.3). For y > 0 we have, from Lemma 3.1, the following identity
Then,
Let us denote
and estimate
Note that the last term goes to zero by the Monotone Convergence Theorem since Ky(x−ξ) → 0 as y → 0 in a monotone way. Now, from either Lemma A.1, Lemma A.2 and Lemma A.3 we get
for α either p or 2p − 2 depending on the range of p. This term again goes to zero as y → 0 by the Monotone Convergence Theorem.
4 Nonlinear Balakrishnan’s formula
The aim of this section is to prove the corresponding Balakrishnan-type formula for p ≠ 2.
Theorem 4.1
Let s ∈ (0, 1) and p ∈ (1, +∞). Assume
where the constant is given by
Notice that, when p = 2 we have that
Hence, we recover the usual Balakrishnan’s formula.
To show (4.1) in Theorem 4.1 we need some preliminary results concerning the representation formulas of (t−Δ)−1 and Δ(t−Δ)−1.
Lemma 4.1
Let f ∈ L1(ℝn)∩ L∞(ℝn). The following representation formulas hold for all t = 0:
(t−Δ)−1f = Kt*f with
where G is the Gaussian Kernel, i.e.,
Δ(t−Δ)−1f = Rt*f − f, where Rt(x) = tKt(x). Moreover,
(4.2)
Proof
First, taking the Fourier transform
Since
The intuition for (b) is clear and it comes from the fact that ∂tG − ΔG=δ0. We can formally write
Hence,
We check (4.2) now. First, we note that
Then,
We are now ready to prove Theorem 4.1.
Proof of Theorem 4.1
We proceed in several steps.
Step 1: We will check here that given f ∈ L∞(ℝn), α<0 and a finite M > 0 we have that
From Lemma 4.1 we have
where we have used the fact that ∥G(τ, ∣·∣)∥L1(ℝn) = 1, which implies
From this observation, we trivially get
Now, given ε >0, we consider the bounded function gε : ℝn × ℝn → ℝ given by gε(x, y) = Φp(u(x) − u(y)) χ∣x−y∣>ε. Note that gε(x, x) = 0. Here we will check that the following identity holds
where
Step 2 (a): First we check that
uniformly in M. In this way, the Dominated convergence theorem we get
To check (4.4), we assume first that p ≤ 2, and use the explicit representation of Δ(t−Δ)−1 given in Lemma 4.1 together with Lemma A.1 to get
We use now the explicit form of Rt to get
It is rather standard to check that the last two integrals are finite. Thus, (4.5) is proved for p ≥ 2. When p ∈ (1, 2) we use Lemma A.2 and Lemma A.3 instead and the proof follows in the same way.
Step 2 (b): We proceed now by direct computations:
For any y ≠ x
We just need to check that C is finite which will ensure we can pass to the limit in the last term of (4.6) (we can also do it in the first by Step 2(a)) to get (4.3) by the Dominated Convergence Theorem since
Let us now compute C explicitly:
Clearly C < +∞, which finishes the proof of Step 2.
Step 3. Lastly, we pass to the limit as ε → 0 in (4.3). The limit of the right hand side term was already shown in Step 1 of the proof of Theorem 2.1. For the left hand side, when p ≥ 2, we have by Lemma A.1 (we can proceed analogously with Lemma A.2 and Lemma A.3 when p ∈ (1, 2)) we compute directly
Clearly the left integral in the last equation goes to 0 as ε → 0 since
which again goes to 0 as ε → 0 since
5 The choice of a suitable constant
The choice of the constant C1 > 0 in (1.6) is indifferent in most cases and can be put to 1, but a precise value C1(n, s, p) can be important in writing some exact formulas or in the limit cases s → 0+, s → 1−, or p → 2. Thus, in the linear case p=2 the natural value is fixed by the assumption that the Fourier transform of the operator is a power multiplier,
However, this choice may not seem to be complete when p≠=2. One might expect the choice of C1 to depend on p in such a way that the expected limits are recovered:
The choice (5.1) satisfies the second limit, but not the first one. We propose the following constant.
Lemma 5.1
Consider the constant
Proof
Consider the operator
We will use the following limit
When ∇ϕ(x) ≠ 0, (5.3) is precisely [17, Theorem 2.8]. The situation when ∇ϕ(x) = 0 is even simpler. The case p = 2 is classical and we omit it (see for example [30]). When p > 2 we rely on the following intermediate step in the proof of [17, Theorem 2.8] (see also the proof of [29, Theorem 6.1]):
Clearly the right hand side term in the equation above is zero when ∇ϕ(x)=0. Finally, if ∇ϕ(x)=0, it is straightforward to see that −Δpϕ(x)=0 for p > 2. This proves (5.3). Then,
where we have used the fact that Γ(1)=1. This proves the first limit in (5.2). For the second limit, we just need to check that C1(n, s, 2) coincides with cn, s given in (5.1). This is just a simple computation
where we have used the fact that
Remark 5.1
With the choice of C1(n, s, p) given by Lemma 5.1 we have that
and
An important fact is that these constants are independent of the dimension. It is also standard to check that
which recover the classical constants for the semigroup formula and the extension problem (see for example [38]).
6 A new spectral-type fractional p-Laplacian on domains
There are many well known applications and interesting remarks of the semigroup formula and the extension problem for the fractional Laplacian. After the results that we have shown in this paper, a number of applications can be naturally proposed in the context of the fractional p-Laplacian. Here is a first application.
Let Ω be a smooth domain (bounded or otherwise). The semigroup representation (1.8) motivates the definition of a new operator using formula (1.11), where we replaced −Δ by −ΔΩ, the Laplacian with homogeneous Dirichlet boundary conditions. It is clear that
This is, to our knowledge, a new example of nonlocal operator when Ω ≠ ℝn. We will show below it does not coincide with
First, we show it is well-defined for functions defined exclusively in Ω, at least when they are smooth. This will allow to consider a boundary value problem with boundary conditions in ∂Ω, without taking the set ℝn\Ω into account.
Theorem 6.1
Let s ∈ (0, 1) and assume that
is well-defined at every x0 ∈ Ω. If p is close to 1, more precisely
Proof
Since Ω is smooth, we can extend u to a
We do this by following the proof done above with
We write
For the first term on the right-hand side we can compute the limit, which is well-defined by our knowledge of
For the other two terms in (6.2), we will show the convergence by checking they are Cauchy sequences. To this effect, we write for k > m > 0 that
We need to show that the two remaining terms give Cauchy sequences, so we take k > m > 0. The second term is easy to estimate
where p and q are bounded by positive and negative powers and C, D > 0. Since e−Dtq(t) ∈ L1(0, 1), the second term of the right-hand side of (6.2) is a Cauchy sequence.
For the last term of (6.2), using the comparison principle we have that
Thus,
Hence, the last term of (6.2) is also a Cauchy sequence. □
Remark 6.1
This operator is indeed different from the usual one on domains, which uses the formula of
This happens because both integrands are nonnegative functions, and
Remark 6.2
We can similarly define operators via the extension problem and Balakrishnan’s formulas. For example,
It is an interesting open problem to check whether these definitions are equivalent. Another formula can be written via the extension.
7 Numerical applications
Among the many possible applications of our representations, we want to comment next on the construction of finite-difference methods for the fractional p-Laplacian. We recall that monotone finite-difference discretizations for the fractional Laplacian (case p = 2) have been established in the papers [21] (in ℝ) and [27] (in ℝn) based on the semigroup representation. The advantage of these types of discretizations is that they have a quadratic error order, independently of the fractional power of the operator (see [27]). This is a great advantage compared to other finite difference discretizations in the literature (see discussion in [27, 28]).
Similar ideas could be used in the case of the fractional p-Laplacian using the results of Theorem 2.1. Let us explain the process at a formal level. Let h > 0 and
The already-classical theory shows that this is an “asymptotically good” approximation of Δ (in the sense that it produces convergent and stable schemes in many settings). In accordance to our representation ideas, we are led to define the following discrete operator
by simply replacing Δ by Δh in (1.8). In this setting,
At least formally, we have an alternative representation of
where
Finally, the change of variables τ=t/h2 allows to write
This process is rigorous when sp<2. However, the weights Kβ, h are not well defined when sp≥2: for example when n=1, we note that
which is only enough to show the convergence of the integral in the mentioned range. However, when sp≥2, it is easy to show that formula (7.2) fails to have a finite value at a point x of global strict maximum of u. An idea to solve this problem is introducing an extra discretization parameter δ > 0 and alternatively define
In this way, all the integral are finite and all the steps are well justified rigorously in the whole range p ∈ (1, ∞) and s ∈ (0, 1). One should make δ → 0 as h → 0 to show that the discretization is consistent in some proper sense.
Remark 7.1
A different finite difference discretization of the from of (7.3) for nonlocal p-Laplacian type operators (in the sense of [1]) has been obtained in [49]. They are based on obtaining quadrature rules for a singular integral definition like (1.1). In both cases the obtained discretizations are monotone, which yields good properties of the associated schemes.
Remark 7.2
Balakrishnan’s formula has been the starting point for one of the most popular numerical schemes for the fractional Laplacian. The scheme combines a quadrature for the time integral and a numerical scheme for a sequence of the spatial operators Δ(t−Δ)−1. One of the main advantages of this idea is the fact that, contrary to others, it is suited for parallel computing. We refer to [13] as the basic reference to the topic.
Remark 7.3
The extension problem has also been successfully used by the numerical analysis community in the context of the fractional Laplacian. We refer the reader to [26] for a finite difference treatment in the context of nonlocal porous medium equations and to [48] for a finite element one in the context of boundary value problems.
8 Other applications
We collect here a series of directions in which our previous ideas can be developed.
8.1 Equivalent representations of the Ws, p(ℝn) seminorm
We can define an equivalent way of defining the (s, p)-energy (1.7). More precisely, consider the Gagliardo semi-norm
It is standard to check that we have the following alternative definition
This formulation was introduced in [54]. We also refer to [32] for a modern reference on the topic in the context of Carnot groups.
The corresponding version with Balakrishnan’s formula, is
We have not found this expression in the literature. A similar formula can be constructed via the extension, but it comes as an awkward product of gradients.
8.2 Spectral (s, p)-Laplacian on manifolds
Let (M, g) be a Riemannian manifold. Some authors (see, e.g. [33]) have used the (s, p)-fractional Laplacian on manifolds given by
where g is a Riemannian metric on M. This a regional-type operator. Following our approach for bounded domain, we propose also the spectral-type operator
where Δg is the Laplace-Beltrami operator on M. The change in sign with respect to the other formulas is due to the convention in Geometry, where Δg is the positive operator.
The idea of dealing with this kind of operators has been considered in the case of the fractional versions of the heat equation and porous medium equations. The extension method was used by Banica et al. [4] for the fractional heat equation on hyperbolic space or non compact manifolds. The spectral approach was used quite recently by Berchio et al. [6] for the fractional porous medium equation posed in hyperbolic space.
8.3 Representation via a general parametric family of operators
Following the philosophy of Section 2 and Section 4 we can think of more general representation formulas for the fractional p-Laplacian. Consider a parametric operator family {Tt}t≥0 defined via convolution with measure kernels K(t, ·), i.e.,
At minimum, we need to assume that
for some finite constant CK ≠ 0. Then, we can expect that the following formula holds
with CT = C1(n, s, p)/CK. These formulas have a particularly nice form if K is self-similar, like for example
so that
In some settings, our proofs of the semigroup and the Balakrishnan’s formulas can be repeated. However, if K changes sign and it is not L1, some estimates are likely more difficult. This can be the case, for example, when working with Tt = (−Δ)k (t − Δ)m.
Remark 8.1
Note that when we use the heat semigroup, Tt = etΔ, we have
Here, γ = 1/2, and we must take α = sp/2. It works for all s and p, see (2.1).
Remark 8.2
Another option is to use a fractional heat semigroup,
Let us sketch the proof. It is well known, see for instance [9, 11], that the fractional semigroup has a kernel K with scaling parameters β = n/2σ and γ = 1/2σ, and a profile F that decays like F(r) ∼ r−(n+2σ) as r → ∞. It follows that we must take α = sp/2σ. In view of the decay of F, the integral for CK is convergent only if sp<2σ, and then the technique applies. On the contrary, the fat tail of F is incompatible with this representation if sp≥2 because then we cannot find a σ<1 such that CK is finite.
9 Final comments and open problems
Intuition behind our results: splitting the formula for the fractional p-Laplacian
An idea behind the representation formula (1.8) is the following simple principle: the fractional p-Laplacian (which is both a nonlinear and nonlocal operator) of a function u:ℝn → ℝ can be formally seen as the composition of a linear nonlocal operator acting on n variables of a certain function v:ℝ2n → ℝ which is built from Φp and u. Indeed, let v(x, y) = Φp(u(y) − u(x)). Notice that v(x, x) = 0. At least formally, for sp < 2, and for convenient functions, we have that
This shows a splitting between the linear part
which does not coincide with
Other representation formulas
For the case p = 2, other representation formulas are known (see, e.g. [38]). It seems likely that some of them can be adapted for the case p ≠ 2, but we have chosen not to deal with that issue in this paper.
Optimal domains of definition for the new formulas
After our proofs of equivalence of the operator definitions when applied to functions with a required regularity, an interesting issue is to extend the domain of definition of the operators, as hinted in Subsection 8.1 or done in reference [38] for the linear case, and then to show that the equivalence still holds, at least in some weak or viscosity version. This is a delicate issue that we have decided to postpone. It could be very important for the possible applications to stationary or evolution problems.
Other operators arising from subordination for nonlinear semigroups
In the paper [22], Cipriani and Grillo proposed another type of nonlinear and nonlocal operators defined via subordination applied directly to nonlinear semigroups. More precisely, given a nonlinear semigroup Tt, they define (up to a normalisation constant)
After our semigroup representation formula (1.8) for the fractional p-Laplacian
We perform now a scaling-based argument which reveals that the operator
From here, it is standard to get
so that
On the other hand, direct computations show that
and
If both operators were the same, we must have p = 2 + (q − 2)s and p = q which is clearly impossible for a given s < 1 unless p = q = 2.
Uniqueness of the extension problem (3.1) for sp≥2
For sp<2 the uniqueness of energy solutions of (3.1) is shown in [19]. However, for sp≥2 the diffusion becomes too singular at y = 0 to work properly in the energy setting. It is an open problem to show in which sense of solution Es, p[f] is the unique solution of the boundary value problem.
Numerical methods for the spectral-type fractional p-Laplacian on domains
Following the ideas of [23], one can implement numerical discretization of the fractional p-Laplacian on bounded domains proposed in Section 6.
Regularity and well-posedness
The extension techniques have been extensively used in the theory of regularity of nonlocal problems involving linear operators, i.e the case p=2 of the theory presented in this paper. We refer for example to the seminal work of Athanasopoulos and Caffarelli [2] in the context of fractional porous medium and Stefan type problems. It has also been used to prove well-posedness and properties of such type of equations (see [24, 25]).
We believe that our theory could also help to develop similar results for equations involving the fractional p-Laplacian.
Appendix A Technical lemmas
Lemma A.1
Let
Proof
By symmetry
Now, since p ≥ 2, we have for some ∣ξ∣ ≤ supx, z ∣u(x) − u(x + z)∣ ≤ ∣∇u∣L∞(ℝd) ∣z∣
and the proof is done.
Lemma A.2
Let
for some constant c depending on p, n and ∣u∣C2(ℝn).
Proof
We follow the proof of Lemma 3.6 in [36] to show that
Changing to polar coordinates and using Lemma 3.5 in [36] we get
where in the last inequality we have used the fact that f(t) = ∣t∣p−2 is a decreasing function for p < 2. Changing back to regular coordinates, we get the desired result.
Lemma A.3
Let
for some constant c depending on p, n and ∣u∣C2(ℝn).
Proof
We follow in the proof of Lema 3.6 [36] to show that
where in the last inequality we have used the fact that f(t) = ∣t∣p−2 is a decreasing function for p < 2.
Remark A.1
If K(z) = ∣z∣−n−spχ∣z∣≤1(z), then Lemma A.3 applies for
that is finite if sp + 2 − 2p < 0, that is
Acknowledgements
F. del Teso was partially supported by PGC2018-094522-B-I00 from the MICINN of the Spanish Government. The work of D. Gómez-Castro and J. L. Vázquez was funded by grant PGC2018-098440-B-I00 from the Spanish Government. D. Gómez-Castro was supported by the Advanced Grant Nonlocal-CPD (Nonlocal PDEs for Complex Particle Dynamics: Phase Transitions, Patterns and Synchronization) of the European Research Council Executive Agency (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 883363). J. L. Vázquez is an Honorary Professor at Univ. Complutense de Madrid. We thank Bruno Volzone and the referees for interesting suggestions.
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