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Publicly Available Published by De Gruyter August 23, 2021

Three representations of the fractional p-Laplacian: Semigroup, extension and Balakrishnan formulas

  • Félix del Teso , David Gómez-Castro EMAIL logo and Juan Luis Vázquez


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

(1.1) (Δ)su(x)=cn,sPVRdu(x)u(y)|xy|n+2sdy,

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

(1.2) J2,s(u)=RdRd|u(x)u(y)|2|xy|n+2sdxdy,

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

(1.3) (Δ)su(x)=1Γ(s)0(etΔ[u](x)u(x))dtt1+s,

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

(1.4) (Δ)su(x)=4sΓ(s)Γ(s)limy0v(x,y)v(x,0)y2s.

(iii) A further representation for the fractional Laplacian was given by Balakrishnan whose formula reads (see [3])

(1.5) (Δ)su(x)=sin(πs)π0(Δ)(tΔ)1[u](x)dtt1s.

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

(1.6) (Δ)psu(x)=C1PVRdΦp(u(x)u(y))|xy|n+spdy,

where p ∈ (1, ∞) and s ∈ (0, 1), we write Φp(t)=∣tp−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)

(1.7) Jp,s(u)=C1pRdRd|u(x)u(y)|p|xy|n+spdxdy,

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, (Δ)psu(x) is well-defined in the pointwise sense for smooth functions Cc(Rn) (assuming also that the gradient does not vanish when p lies in the range p(1,22s) (see, e.g., Section 3 in [36]).

Our main result is the introduction of three equivalent representation formulas for (Δ)ps , which retain the flavour of the linear case. First, we show that the fractional p-Laplacian can be represented by the formula

(1.8) (Δ)psu(x)=C20+etΔ[Φp(u(x)u())](x)dtt1+sp2,

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:

(1.9) (Δ)psu(x)=C3limy0Es,p[Φp(u(x)u())](x,y)ysp,

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

(1.10) (Δ)psu(x)=C40Δ(tΔ)1[Φp(u(x)u())](x)dtt1sp2,

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 (Δ)psu(x) is computed. As a consequence of that, our representation formulas need to solve a single problem with an initial/boundary condition depending on a parameter, i.e. fλ(y) = Φp(λu(y)) where λ = u(x). This fact is very clearly seen in the numerical application, since formula (7.4) does not add any extra complexity with respect its linear counterpart in the computation of the coefficients. Another possible interpretation of our representations is that given a function u : ℝn → ℝ we build an initial/boundary condition F(x, z)≔ Φp(u(x) − u(z)) and solve a problem in R+2n+1 where the operator only acts in n + 1 of the variables. We continue next with some further consequences.

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

(1.11) (ΔΩ)psu(x):=C20+etΔΩ[Φp(u(x)u())](x)dtt1+sp2,

where etΔΩ denotes the heat semigroup in Ω with Dirichlet boundary conditions, i.e. w(x,t):=etΔΩ[f](x) is the solution of


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 (Δ)ps . Furthermore, we suggest a generalisation for manifolds that we do not develop in full detail. Analogous variants can be constructed via (1.9) and (1.10).

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 (Δ)ps treated here. On the other hand, Bjorland-Caffarelli-Figalli introduce in [8, 7] a kind of normalized (game theoretical) fractional infinity Laplacian as well as a fractional p-Laplace operator in non-divergence form (see [44,45] for the local version). They are the nonlocal counterpart of the normalized standard p-Laplacian in the viscosity sense. Finally, Chasseigne and Jakobsen [20] introduced a related non-normalized nonlocal p-Laplacian in the viscosity sense.

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 uCb2(Rn) and x0∈ ℝn. If p(1,22s) assume additionally thatu(x0) ≠ =0. Then we have the following representation of the fractional p-Laplacian

(2.1) (Δ)psu(x0)=C2(n,s,p)0+etΔ[Φp(u(x0)u())](x0)dtt1+sp2,

where C2(n,s,p)=πn2C1(n,s,p)2spΓ(n+sp2) .

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)) χxy∣ > 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 dt/t1+sp2 in the truncated domain (δ, +∞) (we do this to avoid integrability issues around the origin due to the singular behaviour of the measure) to obtain


where we have used Fubini’s theorem to change the order of integration. We use now the change of variables t = ∣xy2/(4τ) to get


Note that, since gε(x, y)=0 if ∣xy∣ < ε, we can consider the above expression to be integrated in ℝnBε(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 eττ1+n+sp2 which is an integrable function in the domain (0, +∞), which yields to the definition of the Γ function. To pass to the limit in the left hand side, we estimate


where we have used either Lemma A.1 or Lemma A.2 with


and the change of variables z=t12r i.e. dz=tn2dr . Then


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) = ∣znspχz∣<ε(z) to get


While in the left hand side we use either Lemma A.1 or Lemma A.2 with K(z)=e|z|24tχ|z|<ε(z) to estimate for t ∈ (0, 1),


and for t = 1,



(2.2) |0+etΔ[Φp(u(x)u())](x)dtt1+sp20etΔ[gε(x,)](x)dtt1+sp2|01|r|<εt12er24|r|pdrdtt1(1s)p2+εmin{1,p1}1dtt1+sp2.

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, ∞),




so by the Dominated Convergence Theorem this term also converges to zero as ε → 0.

Step 2: Now let p(22s,2) . We can use Lemma A.3 to get


so that


which is finite since p1>sp2 since p>22s . That allow us to pass to the limit as δ → 0. The limit as ε → 0 follows also as in Step 1 one using Lemma A.3 instead of Lemma A.1.

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

(3.1) ΔxU(x,y)+1spyUy(x,y)+Uyy(x,y)=0xRn,y>0,U(x,0)=f(x)xRn,

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

(3.2) U(x,y)=RnP(xξ,y)f(ξ)dξ,whereP(x,y)=Γ(n+sp2)πn2Γ(sp2)ysp(|x|2+y2)n+sp2.

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 uCb2(Rn) and x0∈ ℝn. If p(1,22s) assume additionally thatu(x0) ≠ = 0. We have the following representation of the fractional p-Laplacian:

(3.3) (Δ)psu(x0)=C3(n,s,p)limy0Es,p[Φp(u(x0)u())](x0,y)ysp,

where the constant is given by C3(n,s,p)=πn2Γ(sp2)Γ(n+sp2)C1(n,s,p) .

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 fCb(ℝn) then Es, p[f] is a classical solution of (3.1). Furthermore, it can be alternatively represented by

(3.4) Es,p[f](x,y)=ysp2spΓ(sp2)0etΔ[f](x)ey24tdtt1+sp2.


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 ey24t/tα0 as t → 0 for any α ∈ ℝ as long as y > 0.

Second, for, (3.2)


which again is finite since K(ξ):=(|ξ|2+y2)n+2s2 is bounded as long as y > 0 and K(ξ) ⩽ ∣ξ−(n+2s) which is an integrable function in ℝnB1.

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




Let us denote


and estimate


Note that the last term goes to zero by the Monotone Convergence Theorem since Ky(xξ) → 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 uCb2(Rn) and x0∈ ℝn. If p(1,22s) assume additionally thatu(x0) ≠ 0. We have the following representation of the fractional p-Laplacian:

(4.1) (Δ)psu(x0)=C4(n,s,p)0Δ(tΔ)1[Φp(u(x0)u())](x0)dtt1sp2,

where the constant is given by C4(n,s,p)=πn22spΓ(2+sp2)Γ(n+sp2)C1(n,s,p) .

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 fL1(ℝn)∩ L(ℝn). The following representation formulas hold for all t = 0:

  1. (t−Δ)−1f = Kt*f with Kt(x)=0etτG(τ,x)dτ ,

    where G is the Gaussian Kernel, i.e.,

  2. Δ(t−Δ)−1f = Rt*ff, where Rt(x) = tKt(x). Moreover, Rt(x)=tn2W(t12|x|) with W : ℝ+ → ℝ+ given by

    (4.2) W(ρ)=ρ2n(4π)n20eρ2wwn2e14wdw.


First, taking the Fourier transform


Since F[G(t,)](ξ)=e|ξ|2t we recover (a). As shown in [46, Section 3.2], we have


The intuition for (b) is clear and it comes from the fact that ∂tG − ΔG=δ0. We can formally write




We check (4.2) now. First, we note that G(τ,x)=τn2F(τ12|x|) with F : ℝ+ → ℝ+ given by




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 fL(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)) χxy∣>ε. Note that gε(x, x) = 0. Here we will check that the following identity holds

(4.3) 0Δ(tΔ)1[gε(x,)](x)dtt1sp2=C|xy|>εΦp(u(x)u(y))|xy|n+spdy,

where C=2spπn2Γ(1+sp2)Γ(n+sp2)>0 .

Step 2 (a): First we check that

(4.4) 0M|Δ(tΔ)1[gε(x,)](x)|dtt1sp2<+

uniformly in M. In this way, the Dominated convergence theorem we get

(4.5) 0MΔ(tΔ)1[gε(x,)](x)dtt1sp2M0Δ(tΔ)1[gε(x,)](x)dtt1sp2<+.

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:

(4.6) =|xy|>εΦp(u(x)u(y))0MRt(xy)dtt1sp2dy=|xy|>εΦp(u(x)u(y))|xy|n+sp0M|xy|2W(τ12)dτ(τ12)2nspdy.

For any yx


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 fε(t):=|z|<εRt(z)dz0 pointwise as ε → 0 due to the fact that ∥RtL1(ℝn) = 1. For the right integral, we have


which again goes to 0 as ε → 0 since |y|<εt12W(|y|)|y|pdy0 a.e. in t as ε → 0. This finishes the proof.

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, F[(Δ)su]=|ξ|2sF[u] , cf. [51]. Working out the details, the literature points out the value

(5.1) cn,s=22sΓn+2s2πn2|Γ(s)|=s(1s)22sΓn+2s2πn2Γ(2s).

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:

(5.2) (Δ)psϕs1Δpϕ(x)for all p[2,)(Δ)psϕp2+(Δ)sϕ(x)for all s(0,1).

The choice (5.1) satisfies the second limit, but not the first one. We propose the following constant.

Lemma 5.1

Consider the constant


in (1.6). Then, for ϕCb2(Rn) we have that (5.2) holds.


Consider the operator


We will use the following limit

(5.3) (1s)Lps[ϕ](x)s1πn12pΓ(p+12)Γ(n+p2)Δpϕ(x).

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 Γ(1/2)=π and zΓ(z) = Γ(z + 1). This finished the proof.

Remark 5.1

With the choice of C1(n, s, p) given by Lemma 5.1 we have that




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 (ΔΩ)ps coincides with (Δ)ps if Ω = ℝn, and one can even expect some “continuous” dependence on Ω.

This is, to our knowledge, a new example of nonlocal operator when Ω ≠ ℝn. We will show below it does not coincide with (Δ)ps applied to the extension of u by 0 outside Ω. On the other hand, when p=2 we recover the usual Spectral Fractional Laplacian, which is simply the usual spectral power of the Dirichlet Laplacian in a bounded domain as a compact operator; we refer the reader to [18, 52,12] for modern references.

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 uC2(Ω¯) , where Ω is a smooth subdomain onn. If p22s , the following formula

(6.1) (ΔΩ)psu(x0):=C20+etΔΩ[Φp(u(x0)u())](x0)dtt1+sp2

is well-defined at every x0 ∈ Ω. If p is close to 1, more precisely p(1,22s) , we need to assume additionally thatu(x0) ≠ 0.


Since Ω is smooth, we can extend u to a Cc2(Rn) , via the tubular neighbourhood theorem. We will still denote the extended function by u. The time integral in formula (6.1) could be singular at both ends, t = 0, ∞, hence we must show the convergence of the integral at either end. The integral over the interval [1, ∞) is not a problem, since etΔΩfL(Ω)Ceλ1tfL(Ω) where λ1 is the first eigenvalue of the Dirichlet Laplacian. Let us show that there exists


We do this by following the proof done above with (Δ)ps , with suitable modifications. For convenience, we write the semigroup etΔΩ in its kernel form KΩ, and compare with the heat kernel of the whole space Kn. We define g(x, y) = Φp(u(x) − u(y)). Then,


We write

(6.2) 1/m1ΩKΩ(t,x,y)g(x,y)dydtt1+sp2=1/m1RnKRn(t,x,y)g(x,y)dydtt1+sp21/m1RnΩKRn(t,x,y)g(x,y)dydtt1+sp21/m1Ω(KRn(t,x,y)KΩ(t,x,y))g(x,y)dydtt1+sp2.

For the first term on the right-hand side we can compute the limit, which is well-defined by our knowledge of (Δ)ps , notice that by (2.2) in the proof Theorem 2.1 we have


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 eDtq(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




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 (Δ)ps restricted to functions u defined in ℝn such that u = 0 outside Ω (see, for example, [47,57]). Let Ω be fixed and assume without loss of generality that 0 ∈ Ω. Let BR(0) ⊂ Ω and u a smooth function that is positive, radially symmetric, and strictly decreasing along the radius and 0 outside of BR. Then, with the notation of the proof of Theorem 6.1


This happens because both integrands are nonnegative functions, and

KRn(t,x,y)KΩ(t,x,y)0for allx,yΩ.

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 {ei}i=in be standard basis of ℝn. Consider the discrete Laplacian given by

(7.1) Δh[u](x)=i=1nu(x+hei)+u(xhei)2u(x)h2.

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

(7.2) (Δh)ps[u](x):=C20etΔh[Φp(u(x)u())](x)dtt1+sp2,

by simply replacing Δ by Δh in (1.8). In this setting, etΔh[f] represents the uniquely-defined solution of ∂twhw such that w(x, 0)=f(x). We will call it the discrete fractional p-Laplacian.

At least formally, we have an alternative representation of (Δh)ps given in a finite-difference form similar to (7.1). We follow here the presentation of [27]. Let zββ h for β ≔ (β1, …, βn) ∈ ℤn and ψ:ℝn → ℝ. We have that


where G(β,t)=e2ntΠi=1nI|βi|(2t) and Im is the modified Bessel function of first kind with order m ∈ ℕ. From (7.2) we get, at least formally,


Finally, the change of variables τ=t/h2 allows to write

(7.3) (Δh)psu(x)=β0Φpu(x)u(x+zβ)Kβ,hwithKβ,h=C2hsp0G(β,t)dtt1+sp2.

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

(7.4) (Δh,δ)ps:=C2δetΔ[Φp(u(x)u()](x)dtt1+sp2=β0Φpu(x)u(x+zβ)Kβ,h,δ,
(7.5) withKβ,h,δ=C2hspδh2Gh(β,t)dtt1+sp2.

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

0K(t,y)dtt1+α=CK|y|n+spfor all yRn,

for some finite constant CK ≠ 0. Then, we can expect that the following formula holds

(8.1) (Δ)psu(x)=CT0+TtΦp(u(x)u())(x)dtt1+α

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, Tt=et(Δ)σ , with σ ∈ (0, 1). Then the representation (8.1) is correct on the condition that sp<2σ.

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 (Δ)sp2 and the nonlinear part Φp. For sp≥2 one should define the operator


which does not coincide with (Δ)sp2 . Naturally, this very singular integral is only defined for special ψ and x. A key part of our arguments are based on showing that, even though the operator Lsp2 is hardly ever defined, the composition works. The scaling properties of the corresponding kernels and integrals in t or z will be crucial. This splitting idea is also shared by the two other alternative representations (1.9) and (1.10).

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)

(9.1) As[u](x):=0u(x)Tt[u](x)dtt1+s.

After our semigroup representation formula (1.8) for the fractional p-Laplacian (Δ)ps one might wonder if As coincides with (Δ)ps for some semigroup associated to a local operator, as is happens for the case p=2 (the fractional Laplacian). Here the natural candidate is the semigroup Ttq associated to the p-Laplacian operator Δqu ≔ ∇(∣∇ uq−2u) for q > 1, i.e., Ttq[u](x):=w(x,t) , where


We perform now a scaling-based argument which reveals that the operator Asq defined by (9.1) for Ttq is indeed different than the operator (Δ)ps for any choice of p, q ∈ (1, ∞)\{2}. Clearly, for any q ≠ 2 and h > 0 we have


From here, it is standard to get


so that

(9.2) Asq[hu](x)=h1+(q2)sAsq[u](x)andAsq[u(h)](x)=hsqAsq[u](hx).

On the other hand, direct computations show that




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

Appendix A Technical lemmas

Lemma A.1

Let uCb2(Rd) and Φp(t) = ∣tp−2t for p ≥ 2. Let K: ℝd → ℝ+ be such that K(z) = K (−z) and K(z)|z|pdz<+ . Then



By symmetry


Now, since p ≥ 2, we have for some ∣ξ∣ ≤ supx, zu(x) − u(x + z)∣ ≤ ∣∇uL(ℝd)z


and the proof is done.

Lemma A.2

Let uCb2(Rd) and Φp(t) = ∣tp−2t for p ∈ (1, 2). Let K: ℝd → ℝ+ be such that K(z) = κ(∣z∣) and K(z)|z|pdz<+ . Ifu(x) ≠ 0, then


for some constant c depending on p, n anduC2(ℝn).


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) = ∣tp−2 is a decreasing function for p < 2. Changing back to regular coordinates, we get the desired result.

Lemma A.3

Let uCb2(Rd) and Φp(t) = ∣tp−2t for p ∈ (1, 2). Let K: ℝd → ℝ+ be such that K(z) = κ(∣z∣) and K(z)|z|2p2dz<+ . Then


for some constant c depending on p, n anduC2(ℝn).


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) = ∣tp−2 is a decreasing function for p < 2.

Remark A.1

If K(z) = ∣znspχz∣≤1(z), then Lemma A.3 applies for p>22s since


that is finite if sp + 2 − 2p < 0, that is p>22s .


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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Received: 2020-12-18
Published Online: 2021-08-23
Published in Print: 2021-08-26

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