ASYMPTOTIC MEAN VALUE PROPERTIES FOR ...

ASYMPTOTIC MEAN VALUE PROPERTIES FOR FRACTIONAL ANISOTROPIC OPERATORS CLAUDIA BUCUR AND MARCO SQUASSINA Abstract. We obtain an asymptotic representation formula for harmonic functions with respect to a linear anisotropic nonlocal operator. Furthermore we get a Bourgain-Brezis-Mironescu type limit formula for a related class of anisotropic nonlocal norms.

Contents 1. Introduction 2. Preliminary results and notations 3. Viscosity setting 4. Some remarks about the weak setting 5. Asymptotics as s % 1 References

1 3 6 7 8 14

1. Introduction This paper presents an asymptotic mean value property for harmonic functions for a class of anisotropic nonlocal operators. To introduce the argument, we notice that as known from elementary PDEs facts, a C 2 function u : Ω ⊂ Rn → R is harmonic in Ω (i.e. it holds that ∆u = 0 in Ω) if and only if it satisfies the mean value property, that is Z u(x) = − u(y)dy, whenever Br (x) ⊂ Ω. Br (x)

As a matter of fact, this condition can be relaxed to a pointwise formulation by saying that u ∈ C 2 (Ω) satisfies ∆u(x) = 0 at a point x ∈ Ω if and only if Z u(x) = − u(y)dy + o(r2 ), as r → 0. (1.1) Br (x)

This asymptotic formula holds true also in the viscosity sense for any continuous function. A similar property can be proved for quasi-linear elliptic operators such as the p-Laplace operator −∆p u in the asymptotic form, as the radius r of the ball vanishes. More precisely, Manfredi, Parviainen and Rossi proved [18] that, if p ∈ (1, ∞], a continuous function u : Ω ⊂ Rn → R is p-harmonic in Ω in viscosity sense if and only if  2 + nZ p−2  max ϕ + min ϕ + − ϕ(y)dy + o(r2 ), (1.2) ϕ(x) ≥ (≤) 2p + 2n Br (x) p + n Br (x) Br (x) for any ϕ ∈ C 2 such that u − ϕ has a strict minimum (strict maximum for ≤) at x ∈ Ω at the zero level. Notice that formula (1.2) reduces to (1.1) for p = 2. Formula (1.2) holds in the classical sense for smooth functions, at those points x ∈ Ω such that ∇u(x) 6= 0. On the other hand, the 2010 Mathematics Subject Classification. 46E35, 28D20, 82B10, 49A50. Key words and phrases. Mean value formulas, anisotropic fractional operators. The second author is member of Gruppo Nazionale per l’Analisi Matematica, la Probabilit` a e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). 1

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C. BUCUR AND M. SQUASSINA

case p = ∞ offers a counterxample for the validity of (1.2) in the classical sense, since the function |x|4/3 − |y|4/3 is ∞-harmonic in R2 in the viscosity sense but (1.2) fails to hold pointwisely. If p ∈ (1, ∞) and n = 2 Arroyo and LLorente [3] (see also [16]) proved that the characterization holds in the classical sense. The limit case p = 1 was finally investigated in [14]. Since the local (linear and nonlinear) case is well understood, it is natural to wonder about the validity of some kind of asymptotic mean value property in the nonlocal case. As a first approach, we want to investigate this type of property for a nonlocal, linear, anisotropic operator, defined as Z ∞ Z δ(u, x, ρω) (1.3) da(ω) dρ Lu(x) = , ρ1+2s Sn−1 0 where δ(u, x, y) := 2u(x) − u(x − y) − u(x + y). Here, a is a non-negative measure on Sn−1 , finite i.e. Z da ≤ Λ Sn−1

for some real number Λ > 0. We refer to this type of measure as spectral measure, as it is common in the literature. We notice that when the measure a is absolutely continuous with respect to the Lebesgue measure, i.e. when da(ω) = a(ω)dH n−1 (ω) for a suitable, non-negative function a ∈ L1 (Sn−1 ), the operator can be represented (using polar coordinates) as   Z dy y . (1.4) Lu(x) = δ(u, x, y)a |y| |y|n+2s Rn Moreover, if a ≡ 1 then the formula gets more familiar, as we obtain the well-known fractional Laplace operator (see, e.g. [9, 11, 22] and other references therein). We remark also that the operator L is pointwise defined in an open set Ω ⊂ Rn when, for instance, u ∈ C 2s+ε (Ω) ∩ L∞ (Rn ). (Here, C 2s+ε (Ω) denotes C 0,2s+ε (Ω) or C 1,2s+ε−1 (Ω) for a small ε > 0, when 2s + ε < 1, respectively when 2s + ε ≥ 1.) As a matter of fact, the operator (1.3) has been widely studied in the literature, being L the generator of any stable, symmetric Levy process. In particular, regularity issues for harmonic functions of L have been studied in papers like [4–6, 13, 23] (check also other numerous references therein). There, some additional condition are required to the measure, in order to assure regularity. A typical assumption when L is of the form (1.4) is 0 < c ≤ a(y) ≤ C

in Sn−1 ,

or less restrictively in a subset of positive measure Σ ⊂ Sn−1 .

a(y) ≥ c > 0

Furthermore, for instance in [4] the measure needs not to be absolutely continuous with respect to the Lebesgue measure. In [20, 21], the optimal regularity is proved for general operators of the form (1.3), with the “ellipticity” assumption Z |ω · ω|2s da(ω) ≥ λ > 0 inf (1.5) n−1 ω∈S

Sn−1

for some real number λ. We note that the assumptions (1.5) are satisfied by any stable operator with the spectral measure which is n-dimensional (that is, when the measure is not supported on any proper hyperplane of Rn ). We will discuss some details related to this ellipticity requirement

ASYMPTOTIC MEAN VALUE PROPERTIES FOR ANISOTROPIC OPERATORS

3

in Section 4. In this paper, for (1.3) and (1.5), we will adopt a potential theory approach, by using a “mean kernel”, and provide a necessary and sufficient condition for a function to be harmonic for L, in the viscosity sense. To be more precise, we denote for some r > 0 Z ∞ Z u(x + ρω) + u(x − ρω) dρ da(ω) Mrs u(x) := c(n, s, a)r2s , (ρ2 − r2 )s ρ r Sn−1 with Z −1 sin πs da . c(n, s, a) := π Sn−1 The following is the main result of the paper. Theorem 1.1. Let Ω ⊂ Rn be an open set and u ∈ L∞ (Rn ). The asymptotic expansion u(x) = Mrs u(x) + O(r2 ),

(1.6)

as r → 0

holds for all x ∈ Ω in the viscosity sense if and only if Lu(x) = 0 in the viscosity sense. We point out here the paper [12], where the author studies a general type of nonlocal operators defined by means of mean value kernels. Furthermore, the readers can check [1, 8, 15] for details on the mean kernel and the mean value property for the fractional Laplacian. As further results, we provide some asymptotics for s % 1 of the operator L and of the mean value Mrs . We also prove a Bourgain-Brezis-Mironescu type of formula, for a nonlocal norm related to the operator L. In fact, as s % 1, we obtain the “integer”, local counterpart of the objects under study. This paper is organized as follows: in the next section we introduce some notations and some preliminary results. Section 3 deals with the viscosity setting and with the proof of Theorem 1.1. In Section 4 we make some remarks about the weak setting. In the last Section 5 we study the asymptotic behavior as s % 1 of our fractional operators and prove a Bourgain-Brezis-Mironescu type of formula. 2. Preliminary results and notations Notations We use the following notations throughout this paper. • For some r > 0 and any x ∈ Rn Br (x) := {y ∈ Rn |x − y| < r},

Br := Br (0).

Sn−1 = ∂B1 . • For any x > 0, the Gamma function is (see [2], Chapter 6): Z ∞ Γ(x) = tx−1 e−t dt. 0

• For any x, y > 0, the Beta function is (see [2], Chapter 6): Z ∞ tx−1 β(x, y) = dt. (t + 1)x+y 0

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C. BUCUR AND M. SQUASSINA

We remark as a first thing the following integral indentity. Lemma 2.1. For any r > 0 2 sin πs 2s r π

Z

∞

r

(ρ2

dρ = 1. − r 2 )s ρ

Proof. Changing coordinates, we get that Z ∞ Z dρ dt 1 ∞ β(1 − s, s) π r2s = = = , 2 2 s s (ρ − r ) ρ 2 0 t (t + 1) 2 2 sin(πs) r where we have used formulas 6.2.2 and 6.1.17 in [2].



We obtain the asymptotic mean value property for smooth functions, as follows. Theorem 2.2. Let u ∈ C 2 (Ω) ∩ L∞ (Rn ). Then u(x) = Mrs u(x) + c(n, s, a)r2s Lu(x) + O(r2 ) as r → 0. Proof. We fix x ∈ Ω and δ > 0 such that B2δ (x) ⊂ Ω. For any 0 < r < δ, by Lemma 2.1 we have that Z ∞ Z 2u(x) − Mrs u(x) dρ da(ω) 2 u(x) − Mrs u(x) = c(n, s, a)r2s 2 )s ρ (ρ − r n−1 Zr ∞ ZS δ(u, x, ρω) = c(n, s, a)r2s dρ da(ω) 2 (ρ − r2 )s ρ n−1 Z ∞r Z S δ(u, x, rρω) = c(n, s, a) dρ da(ω) 2 . (ρ − 1)s ρ 1 Sn−1 Notice that 1 < δ/r, so we write Z ∞ Z Z δ Z r u(x) − Mrs u(x) δ(u, x, rρω) δ(u, x, rρω) dρ da(ω) 2 = + dρ da(ω) 2 s δ c(n, s, a) (ρ − 1) ρ (ρ − 1)s ρ (2.1) Sn−1 1 Sn−1 r

=: I1 + I2 . We have that

∞

Z I1 =

Z dρ

δ r

da(ω) Sn−1

δ(u, x, rρω) 1 s .  1+2s ρ 1− 1 ρ2

Denote by 1  r ∈ 0, ρ δ and for r small enough with a Taylor expansion we have that t :=

(1 − t2 )−s = 1 + st2 + o(t2 ). So

Z ∞ Z δ(u, x, rρω) δ(u, x, rρω) dρ da(ω) +s dρ da(ω) I1 = 1+2s δ δ ρ ρ3+2s Sn−1 Sn−1 r r Z ∞ Z
+ dρ da(ω)δ(u, x, rρω)o ρ−3−2s . Z

∞

δ r

Z

Sn−1

ASYMPTOTIC MEAN VALUE PROPERTIES FOR ANISOTROPIC OPERATORS

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Notice that Z Z Z ∞ ∞ Z δ(u, x, rρω) dρ dρ da(ω) da ≤ 4kukL∞ (Rn ) 3+2s 3+2s δ δ ρ ρ Sn−1 Sn−1 r

r

≤ 2ΛkukL∞ (Rn ) It follows that

∞

Z I1 =

Z dρ

δ r

On the other hand we write Z I2 =

δ r

Sn−1

Z da(ω)

dρ Sn−1

1

Z + 1

da(ω)

δ r

r2+2s δ −2−2s 1−s

.

δ(u, x, rρω) + O(r2+2s ). ρ1+2s

δ(u, x, rρω) ρ1+2s

Z

δ(u, x, rρω) da(ω) dρ ρ1+2s Sn−1



 ρ2s −1 . (ρ2 − 1)s

Then putting together I1 and I2 into (2.1) we have that Z ∞ Z δ(u, x, rρω) u(x) − Mrs u(x) = dρ da(ω) c(n, s, a) ρ1+2s n−1 1 S   Z δ Z r δ(u, x, rρω) ρ2s + dρ da(ω) − 1 + O(r2+2s ) ρ1+2s (ρ2 − 1)s 1 Sn−1 Z ∞ Z δ(u, x, rρω) dρ da(ω) = ρ1+2s n−1 (2.2) S 0   Z Z δ r δ(u, x, rρω) ρ2s dρ da(ω) + −1 ρ1+2s (ρ2 − 1)s 1 Sn−1 Z 1 Z δ(u, x, rρω) − dρ da(ω) + O(r2+2s ) ρ1+2s 0 Sn−1 =: r2s Lu(x) + J + O(r2+2s ). Recalling that u ∈ C 2 (Ω), both for ρ ∈ (1, δ/r) and for ρ ∈ (0, 1) we have that |δ(u, x, rρω)| ≤ r2 ρ2 kukC 2 (Bδ (x)) . We thus obtain |J | ≤ kukC 2 (Ω) r =C

2

Z da(ω)

Z

Sn−1 2s O(r ))

r2 (1 + 2(1 − s)

The last line follows since  Z 1−2s ρ

1

δ r

1−2s

dρ ρ



  Z 1 ρ2s 1−2s −1 + dρ ρ (ρ2 − 1)s 0

.

 ρ2s ρ2−2s − (ρ2 − 1)1−s − 1 dρ = − . (ρ2 − 1)s 2 − 2s

Hence in (2.2) we finally get that u(x) − Mrs u(x) = c(n, s, a)r2s Lu(x) + O(r2 ). This concludes the proof of the theorem.



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C. BUCUR AND M. SQUASSINA

3. Viscosity setting We begin by giving the definitions for the viscosity setting. First of all (as in [10]), we define the notion of viscosity solution. Definition 3.1. A function u ∈ L∞ (Rn ), lower (upper) semi-continuous in Ω is a viscosity supersolution (subsolution) to Lu = 0,

and we write

Lu ≤ (≥) 0

if for every x ∈ Ω, any neighborhood U = U (x) ⊂ Ω and any ϕ ∈ C 2 (U ) such that ϕ(x) = u(x) ϕ(y) < (>) u(y),

for any y ∈ U \ {x},

if we let  (3.1)

v=

ϕ,

in U

u,

in Rn \ U

then Lv(x) ≥ (≤) 0. A viscosity solution of Lu = 0 is a (continuous) function that is both a subsolution and a supersolution. Definition 3.2. A function u ∈ L∞ (Rn ), lower (upper) semi-continuous in Ω, verifies for any x∈Ω u(x) = Mrs u(x) + O(r2 ) as r → 0, in a viscosity sense, if for any neighborhood U = U (x) ⊂ Ω and any ϕ ∈ C 2 (U ) such that ϕ(x) = u(x) ϕ(y) < (>) u(y),

for any y ∈ U \ {x},

if we let  v=

ϕ,

in U

u,

in Rn \ U

then (3.2)

v(x) ≥ (≤) Mrs v(x) + O(r2 ).

We can now prove the main theorem of this paper. Proof of Theorem 1.1. Let x ∈ Ω and any R > 0 be such that BR (x) ⊂ Ω. Let ϕ ∈ C 2 (BR (x)) be such that ϕ(x) = u(x) ϕ(y) < u(y), We let v be defined as in (3.1), hence v ∈ (3.3)

for any y ∈ BR (x) \ {x}. C 2 (B

R (x))

∩ L∞ (Rn ). By Theorem 2.2 we have that

v(x) = Mrs v(x) + c(n, s, a)r2s Lv(x) + O(r2 ).

We prove at first that if u satisfies (1.6) in the viscosity sense given by Definition 3.2 then u is a supersolution of Lu(x) = 0 in the viscosity sense. Since v(x) ≥ Mrs v(x) + O(r2 ) dividing by r2s in (3.3) and sending r → 0, it follows that Lv(x) ≥ 0.

ASYMPTOTIC MEAN VALUE PROPERTIES FOR ANISOTROPIC OPERATORS

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At the same manner, one proves that u is a subsolution of Lu(x) = 0 in the viscosity sense. In order to prove the other implication, if u is a supersolution, one has from (3.3) that v(x) − Mrs v(x) ≥ 0, r2s

lim sup r→0

hence (3.2). In the same way, one gets the conclusion when u is a subsolution.



4. Some remarks about the weak setting In this section we consider the weak setting, and in particular provide the condition for a weak solution to be a pointwise, and a viscosity solution. In this sense, Theorem 1.1 applies also to weak solutions. Let us now define some new spaces by setting 1 Z Z 2 2 da(ω) (∇u(x) · ω) , kukHa1 (Rn ) := [u]Ha1 (Rn ) + kukL2 (Rn ) , [u]Ha1 (Rn ) := dx Sn−1

Rn

 Ha1 (Rn ) := u ∈ L2 (Rn ) [u]Ha1 (Rn ) < +∞ ,

k·kH 1 (Rn )

1 Ha,0 (Rn ) := Cc∞ (Rn )

a

,

and (u(x) − u(x + ρω))2 dx dρ da(ω) |ρ|1+2s Rn R Sn−1

Z [u]Has (Rn ) :=

Z

Z

!1

2

.

Taking into account that 1 2 (v · ω) da(ω) ,

Z

2

kvk := Sn−1

v ∈ Rn ,

is a norm in Rn , it is readily seen that Ha1 (Rn ) is a Banach space. We take the operator L that satisfies the ellipticity assumption (1.5) and we consider weak solutions of the equation Lu = 0 in Ω. In particular, we consider u ∈ L∞ (Rn ) of finite energy, i.e. such that [u]Has (Rn ) < ∞

(4.1) that satisfies Z



u(x) − u(x + ρω) (ϕ(x) − ϕ(x + ρω) dx dρ da(ω) =0 |ρ|1+2s R Sn−1 Z

Z

Rn ∞ Cc (Ω).

for any ϕ ∈ We notice that, from [20, Theorem 1.1, b)] we have that if u ∈ L∞ (Rn ) ∩ C α (Rn ) for some α > 0, we get that u ∈ C α+2s (Br (p)) for all r > 0 and p ∈ Ω for which B2r (p) ⊂ Ω. As remarked in [20], one cannot remove the hypothesis that u ∈ C α (Rn ), in order to obtain the C 2s+α regularity of u. So, in this way, a weak solution of Lu = 0 in Ω is a pointwise solution and a viscosity solution in say, Ω0 , taking [ Ω0 := Br (p). p∈Ω,r>0 B2r (p)⊂Ω

Notice that when Ω has C 2 boundary, Ω0 can be taken as any compactly contained subset of Ω. We have the next result.

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C. BUCUR AND M. SQUASSINA

Theorem 4.1. Let α > 0 be such that α + 2s is not an integer. Let u ∈ L∞ (Rn ) ∩ C α (Rn ) be of finite energy, and be a weak solution of Lu = 0 in Ω. Then u ∈ C 2α+s (Ω0 ) is a pointwise and a viscosity solution of Lu = 0 in Ω0 . 0

Proof. We recall that, as just noticed, u ∈ C 2s+α (Ω ). Hence Lu(x) is well defined for any x ∈ Ω0 . We take any ϕ ∈ Cc∞ (Ω0 ) and since u is a weak solution, we have that Z Z Z (u(x) − u(x + ρω)) (ϕ(x) − ϕ(x + ρω)) 0= dx dρ da(ω) |ρ|1+2s n n−1 ZR ZS ZR (2u(x) − u(x + ρω) − u(x − ρω)) ϕ(x) da(ω) dx dρ = |ρ|1+2s n−1 n S R R with a change of variables, given the smoothness of u. Thus pointwise in Ω0 Z Z 2u(x) − u(x + ρω) − u(x − ρω) 0= dρ da(ω) = 2Lu(x). |ρ|1+2s R Sn−1 Furthermore, consider v that touches u at x0 ∈ Ω0 from below (as defined in (3.1)) i.e. u(x0 ) = v(x0 ), with v − u ≤ 0 in Rn . Then Z ∞ Z 2v(x0 ) − v(x0 + ρω) − v(x0 − ρω) Lv(x0 ) = dρ da(ω) ≥ Lu(x0 ) = 0. |ρ|1+2s n−1 0 S This proves that u is a supersolution (in the same way, one can prove that u is a subsolution), and concludes the proof of the Theorem.  5. Asymptotics as s % 1 In this section we provide some asymptotic properties on the operator L, the mean value defined by Mrs and the semi-norm in (4.1). We study their limit behavior as s approaches the upper value 1. Indeed, re-normalizing (multiplying by (1 − s)) and sending s % 1, we obtain the local counterpart of the operators under study. It is interesting in our opinion, from this point of view, to understand what is the influence of the non symmetric measure da in the limit, with respect to having the dH n−1 measure on the hypersphere. We begin by showing the following. Proposition 5.1. Let u ∈ C 2 (Ω) ∩ L∞ (Rn ). Then for all x ∈ Ω  n Z 1 X 2 da(ω)ωi ωj ∂ij u(x). lim (1 − s)Lu(x) = − s%1 2 n−1 S i,j=1

Proof. We fix x ∈ Ω. By a Taylor expansion, we have that for every ρ > 0 (such that Bρ (x) ⊂ Ω) and every ω ∈ Sn−1 , there exists h := h(ρ, ω), h := h(ρ, ω) ∈ [0, ρ] such that δ(u, x, ρω) = −

ρ2 2 ρ2 hD u(x + hω)ω, ωi − hD2 u(x − hω)ω, ωi. 2 2

Since u ∈ C 2 (Ω), we have that for any ε > 0 there exists r := r(ε) > 0 such that

 (D2 u(x + hω) − D2 u(x))ω, ω ≤ D2 u(x + hω) − D2 u(x) |ω|2 < ε, (5.1) whenever |h| = |hω| ≤ ρ < r.

ASYMPTOTIC MEAN VALUE PROPERTIES FOR ANISOTROPIC OPERATORS

9

Fixing an arbitrary ε and taking the corresponding r := r(ε), we write Z r Z Z ∞ Z δ(u, xρω) δ(u, xρω) Lu(x) = dρ da(ω) 1+2s + dρ da(ω) 1+2s ρ ρ n−1 n−1 0 r S Z rS Z

 1 da(ω)ρ1−2s D2 u(x + hω)ω, ω dρ = − 2 0 n−1 ZS Z

 1 r − da(ω)ρ1−2s D2 u(x − hω)ω, ω dρ 2 n−1 Z ∞0 Z S δ(u, xρω) da(ω) 1+2s dρ + ρ n−1 r  S
1 1 2 = : − Ir,s + Ir,s + Jr,s . 2 Now notice that Z r Z D E
1 Ir,s = dρ da(ω) D2 u(x + hω) − D2 u(x) ω, ω ρ1−2s n−1 Z0 r ZS + dρ da(ω)hD2 u(x)ω, ωiρ1−2s . 0

Sn−1

By using (5.1) we notice that Z r Z Z D E
2 2 1−2s ≤ dρ da(ω) D u(x + hω) − D u(x) ω, ω ρ n−1 0

≤ εΛ

r2−2s . 2(1 − s)

r

dρ ρ1−2s

0

i,j=1

da(ω)ερ1−2s

0

On the other hand, we get that Z r Z Z n X 2 dρ da(ω)hD2 u(x)ω, ωiρ1−2s = ∂ij u(x) Sn−1

Z dρ Sn−1

S

0

r

Z da(ω)ωi ωj Sn−1

Z n r2−2s X 2 ∂ij u(x) da(ω)ωi ωj = 2(1 − s) Sn−1

(5.2)

=

r2−2s 2(1 − s)

i,j=1 n X

2 mij ∂ij u(x),

i,j=1

using the notation Z mij =

ωi ωj da(ω). Sn−1

Multiplying by (1 − s), letting s % 1 we get that n 1 X 1 2 lim (1 − s)Ir,s = mij ∂ij u(x) + O(ε). s%1 2 i,j=1

2 . Notice also that for s close to 1 (hence when In the same way, one gets the same limit for Ir,s for instance s > 1/2) 2r−2s kukL∞ (Rn ) Λ |Jr,s | ≤ . s Thus we obtain lim (1 − s)Jr,s = 0. s%1

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C. BUCUR AND M. SQUASSINA

Using the arbitrariness of ε, it follows that lim (1 − s)Lu(x) = −

s%1

n 1 X 2 mij ∂ij u(x) 2 i,j=1

hence the conclusion.



Remark 5.2. R Notice that the matrix associated to the local operator, given by the constant coefficients Sn−1 da(ω)ωi ωj , is symmetric. We observe then that the local operator that we have obtained is the classical Laplacian, up to a change of coordinates, provided that the matrix is also positive definite. Furthermore, we have the next result. Proposition 5.3. Let u ∈ C 1 (Ω) ∩ L∞ (Rn ). For all x ∈ Ω and any r > 0 with B2r (x) ⊂ Ω Z −1 Z 1 s da lim Mr u(x) = da(ω) (u(x − rω) + u(x + rω)) . s%1 2 Sn−1 Sn−1 Proof. We fix ε ∈ (0, 1), which we will take arbitrarily small in the sequel. We have that Z ∞ Z u(x + ρω) + u(x − ρω) Mrs u(x) 2s =r dρ da(ω) c(n, s, a) (ρ2 − r2 )s ρ n−1 Z ∞r Z S u(x + rρω) + u(x − rρω) = dρ da(ω) (ρ2 − 1)s ρ n−1 Z1 ∞ ZS u(x + rρω) + u(x − rρω) (5.3) = dρ da(ω) (ρ2 − 1)s ρ 1+ε Sn−1 Z 1+ε Z u(x + rρω) + u(x − rρω) + dρ da(ω) (ρ2 − 1)s ρ n−1 1 S =: I1 + I2 . Now

2

|u(x + rρω) + u(x − rρω)| (ρ2 − 1)s ρ 1+ε Sn−1 Z ∞ Z |u(x + rρω) + u(x − rρω)| + dρ da(ω) (ρ2 − 1)s ρ 2 Sn−1 Z Z 2 dρ ≤ 2kukC(B2r (x)) da s s Sn−1 1+ε (ρ − 1) (ρ + 1) ρ Z Z ∞ dρ + 2kukL∞ (Rn ) da 2 (ρ − 1)s ρ Sn−1 2 Z 2 Z ∞ 2ΛkukC(B2r (x)) dρ dρ ≤ + 4ΛkukL∞ (Rn ) s s 1+2s (1 + ε)(2 + ε) 1+ε (ρ − 1) ρ 2 Z

|I1 | ≤

≤

Z

dρ

da(ω)

2ΛkukC(B2r (x)) (1 − ε1−s ) 4ΛkukL∞ (Rn ) + . 1−s s

Notice that since c(n, s, a) lim = s%1 1 − s

Z

−1 da

Sn−1

we obtain lim c(n, s, a)I1 = 0.

s→1

ASYMPTOTIC MEAN VALUE PROPERTIES FOR ANISOTROPIC OPERATORS

11

On the other hand, integrating by parts, we get that Z 1+ε Z 1+ε ε1−s u (x − r(1 + ε)ω) 1 u(x − rρω) d u(x − rρω) dρ 2 dρ(ρ − 1)1−s = − . s s (ρ − 1) ρ (1 − s)(ε + 2) (1 + ε) 1 − s 1 dρ (ρ + 1)s ρ 1 We have that

hence

d u(x − rρω) dρ (ρ + 1)s ρ ≤ c(r)kukC 1 (B2r (x)) , 1+ε

u(x − rρω) ε1−s u (x − r(1 + ε)ω) dρ 2 − ≤ (ρ − 1)s ρ (1 − s)(ε + 2)s (1 + ε) 1 In the same way, we get that Z 1+ε 1−s u (x + r(1 + ε)ω) ε u(x + rρω) ≤ − dρ 2 (ρ − 1)s ρ (1 − s)(ε + 2)s (1 + ε) 1 Z

It follows that Z ε1−s I2 − (1 − s)(ε + 2)s (1 + ε)

ε2−s c(r)kukC 1 (B2r (x)) . 1−s ε2−s c(r)kukC 1 (B2r (x)) . 1−s


da(ω) u (x − r(1 + ε)ω) + u (x + r(1 + ε)ω) n−1

S

ε2−s ≤ c(r)ΛkukC 1 (B2r (x)) . 1−s Multiplying by c(n, s, a) and sending s % 1 we get that
−1 Z R
Sn−1 da lim c(n, s, a)I2 = da(ω) u(x − r(1 + ε)ω) + u(x + r(1 + ε)ω) + O(ε). s%1 (ε + 2)(ε + 1) Sn−1 For ε → 0 we get that lim c(n, s, a)I2 =

s%1

1 2

Z

−1 Z da

Sn−1


da(ω) u(x − rω) + u(x + rω) .

Sn−1

So putting together the limits involving I1 , I2 into (5.3) we obtain the conclusion.



We use now the norms introduced at the beginning of Section 4. We have the next inequality. Proposition 5.4. Let u ∈ Ha1 (Rn ). Then there exists C > 0 independent of s ∈ (1/2, 1) with (1 − s)[u]2Has (Rn ) ≤ Ckuk2Ha1 (Rn ) . Proof. We have that Z

Z dx

Rn

Sn−1

da(ω) (u(x) − u(x + ρω))2 ≤ ρ2 [u]2Ha1 (Rn ) .

Indeed, for all ρ ∈ R, we have Z Z Z 2 2 dx da(ω) (u(x) − u(x + ρω)) ≤ ρ Rn

Sn−1

dx

≤ρ

2

≤ ρ2

Z Rn Z 1

=ρ

Sn−1

Z dx Z dt

Z

Sn−1

Rn 2 [u]Ha1 (Rn ) . 0

2

∇u(x + tρω) · ω dt

da(ω)

Rn

da(ω) Z

dx Sn−1

2

1

Z

Z

0 1

dt (∇u(x + tρω) · ω)2

0

da(ω) (∇u(x + tρω) · ω)2

12

C. BUCUR AND M. SQUASSINA

Therefore (1 −

s)kuk2Has (Rn )

Z

1

ρ1−2s dρ 0 Z Z Z ∞ −1−2s da(ω)(u(x) − u(x + ρω))2 dx dρρ + (1 − s) Sn−1 Rn 1 Z Z Z ∞ −1−2s dx da(ω)(u(x) − u(x − ρω))2 + (1 − s) dρρ s)[u]2Ha1 (Rn )

≤ 2(1 −

≤

1 C([u]2Ha1 (Rn ) +

Rn 2 kukL2 (Rn ) ),

Sn−1

for some positive constant C.



In what follows, we prove a Bourgain-Brezis-Mironescu type property [7] for anisotropic norms. A different type of anisotropicity in the formula was recently investigated in [19] and in [17]. 1 (Rn ). Then, we have the formula Proposition 5.5 (BBM type formula). Let u ∈ Ha,0

lim (1 − s)[u]2Has (Rn ) = [u]2Ha1 (Rn ) .

(5.4)

s→1

Proof. We prove (5.4) first for any u ∈ Cc1 (Rn ). We write
2 u(x) − u(x + ρω) dx dρ da(ω) ρ1+2s Rn R Sn−1
2 Z Z ∞ Z u(x) − u(x + ρω) = dx dρ da(ω) ρ1+2s Rn 0 Sn−1
2 Z Z ∞ Z u(x) − u(x − ρω) . + dx dρ da(ω) ρ1+2s Rn 0 Sn−1 Z

(5.5)

Z

Z

Since u ∈ C 1 by the mean value theorem, there exist h := h(ρ, ω), h := h(ρ, ω) ∈ [0, ρ] such that u(x + ρω) − u(x) = ρω · ∇u(x + hω)

and

u(x − ρω) − u(x) = − ρω · ∇u(x + hω). Furthermore, for any ε > 0 there exists r := r(ε) > 0 such that (5.6)

|∇u(x + hω) − ∇u(x)| < ε

whenever |h| = |hω| < ρ < r.

We fix ε > 0 (to be taken arbitrarily small in the sequel) and consider the correspondent r := r(ε). We then write
2 Z Z ∞ Z u(x) − u(x + ρω) dx dρ da(ω) ρ1+2s Rn 0 Sn−1 Z Z r Z
2 = dx dρ da(ω)ρ1−2s ∇u(x + hω) · ω Rn

Sn−1

0

Z

Z

+

dx Rn

=:

1 Ir,s

+

2 Ir,s .

r

∞

u(x) − u(x + ρω) dρ da(ω) ρ1+2s Sn−1 Z


2

ASYMPTOTIC MEAN VALUE PROPERTIES FOR ANISOTROPIC OPERATORS

13

Notice that 1 Ir,s

Z

Z

dρρ

dx

= n

Z0 r

ZR + =

r

dx Rn 1 Jr,s +

1−2s

Z da(ω) ∇u(x + hω) · ω


2

− ∇u(x) · ω


2

n−1

dρρ1−2s

0 2 Jr,s .

ZS

da(ω) ∇u(x) · ω


2

Sn−1

From (5.6), we have that
2
2 ∇u(x + hω) · ω − ∇u(x) · ω

≤ ∇u(x + hω) − ∇u(x) · ω ∇u(x + hω) − ∇u(x) · ω ≤ 2εkukC 1 (Rn ) . Therefore, for some compact set K ⊂ Rn independent of ε, there holds 2−2s 1 Jr,s ≤ εΛkukC 1 (Rn ) r |K|. 2(1 − s) Also

Z Z r2−2s dx da(ω)(∇u(x) · ω)2 . 2(1 − s) Rn n−1 S It follows that Z Z 1 1 lim (1 − s)Ir,s = dx da(ω)(∇u(x) · ω)2 + O(ε). s→1 2 Rn Sn−1 Furthermore we get that Z ∞ r2s kuk2L2 (Rn ) Λ 2 −1−2s Ir,s ≤ 2kuk2 2 n Λ , ρ dρ = L (R ) s r hence 2 lim (1 − s)Ir,s = 0. 2 Jr,s =

s→1

We finally get that Z Z lim (1 − s) dx s→1

Rn

0

∞

u(x) − u(x + ρω) dρ da(ω) ρ1+2s Sn−1 Z


2 =

1 2

Z

da(ω)(∇u(x) · ω)2 + O(ε).

Sn−1

We obtain the same limit for the second term in (5.5) and get (5.4) for u ∈ Cc1 (Rn ) by sending 1 (Rn ). There exists {u } ∈ C ∞ (Rn ) such that ε → 0. Let now u ∈ Ha,0 j j c ku − uj kHa1 (Rn ) → 0

as j → ∞.

Then, according to Proposition 5.4 we have that
2 (1 − s) [u]Has (Rn ) − [uj ]Has (Rn ) ≤ (1 − s)[u − uj ]2Has (Rn ) ≤ Cku − uj k2Ha1 (Rn ) → 0 1 (Rn ) immediately follows. The conclusion (5.4) for u ∈ Ha,0

as j → 0. 

Remark 5.6. If da = dH n−1 , the left-hand side of the formula in Proposition 5.4 boils down to
2 Z Z ∞ Z u(x) − u(x + ρω) lim (1 − s) dx dρ dH n−1 (ω) , s→1 |ρ|1+2s Rn 0 Sn−1 while the right-hand side to Z Z Z Qn,2 1 dx dH n−1 (ω)(ω · ∇u(x))2 = |∇u(x)|2 dx 2 Rn 2 n−1 S Ω

14

C. BUCUR AND M. SQUASSINA

where Z

|σ · ω|2 dH n−1 (ω)

Qn,2 = Sn−1

for some σ ∈ Sn−1 . This is consistent with the usual Brezis-Bourgain-Mironescu formula (see [7]).

References [1] Nicola Abatangelo. Large S-harmonic functions and boundary blow-up solutions for the fractional Laplacian. Discrete Contin. Dyn. Syst., 35(12):5555–5607, 2015. 3 [2] Milton Abramowitz and Irene A. Stegun, editors. Handbook of mathematical functions with formulas, graphs, and mathematical tables. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York; National Bureau of Standards, Washington, DC, 1984. Reprint of the 1972 edition, Selected Government Publications. 3, 4 ´ [3] Angel Arroyo and Jos´e G. Llorente. On the asymptotic mean value property for planar p-harmonic functions. Proc. Amer. Math. Soc., 144(9):3859–3868, 2016. 2 [4] Richard F. Bass and Zhen-Qing Chen. Regularity of harmonic functions for a class of singular stable-like processes. Math. Z., 266(3):489–503, 2010. 2 [5] Richard F. Bass and Moritz Kassmann. H¨ older continuity of harmonic functions with respect to operators of variable order. Comm. Partial Differential Equations, 30(7-9):1249–1259, 2005. 2 [6] Krzysztof Bogdan and Pawel Sztonyk. Estimates of the potential kernel and Harnack’s inequality for the anisotropic fractional Laplacian. Studia Math., 181(2):101–123, 2007. 2 [7] Jean Bourgain, Haim Brezis, and Petru Mironescu. Another look at Sobolev spaces. In Optimal control and partial differential equations, pages 439–455. IOS, Amsterdam, 2001. 12, 14 [8] Claudia Bucur. Some observations on the Green function for the ball in the fractional Laplace framework. Communications on Pure and Applied Analysis, 15(2):657–699, 2016. 3 [9] Claudia Bucur and Enrico Valdinoci. Nonlocal diffusion and applications. arXiv preprint arXiv:1504.08292, 2015. Accepted for Publication for the Springer Series “Lecture Notes of the Unione Matematica Italiana”. 2 [10] Luis Caffarelli and Luis Silvestre. Regularity theory for fully nonlinear integro-differential equations. Communications on Pure and Applied Mathematics, 62(5):597–638, 2009. 6 [11] Eleonora Di Nezza, Giampiero Palatucci, and Enrico Valdinoci. Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math., 136(5):521–573, 2012. 2 [12] Fausto Ferrari. Mean value properties of fractional second order operators. Commun. Pure Appl. Anal., 14(1):83–106, 2015. 3 [13] Moritz Kassmann, Marcus Rang, and Russell W Schwab. Integro-differential equations with nonlinear directional dependence. Indiana Univ. Math. J, 63(5):1467–1498, 2014. 2 [14] Bernd Kawohl, Juan Manfredi, and Mikko Parviainen. Solutions of nonlinear PDEs in the sense of averages. J. Math. Pures Appl. (9), 97(2):173–188, 2012. 2 [15] Naum S. Landkof. Foundations of modern potential theory. Springer-Verlag, New York-Heidelberg, 1972. Translated from the Russian by A. P. Doohovskoy, Die Grundlehren der mathematischen Wissenschaften, Band 180. 3 [16] Peter Lindqvist and Juan Manfredi. On the mean value property for the p-Laplace equation in the plane. Proc. Amer. Math. Soc., 144(1):143–149, 2016. 2 [17] Monica Ludwig. Anisotropic fractional sobolev norms. Adv. Math., 252:150–157, 2014. 12 [18] Juan J. Manfredi, Mikko Parviainen, and Julio D. Rossi. An asymptotic mean value characterization for p-harmonic functions. Proc. Amer. Math. Soc., 138(3):881–889, 2010. 1 [19] Hoai-Minh Nguyen and Marco Squassina. On anisotropic sobolev spaces. Preprint, 2017. 12 [20] Xavier Ros-Oton and Joaquim Serra. Regularity theory for general stable operators. J. Differential Equations, 260(12):8675–8715, 2016. 2, 7 [21] Xavier Ros-Oton and Enrico Valdinoci. The Dirichlet problem for nonlocal operators with singular kernels: convex and nonconvex domains. Advances in Mathematics, 288:732–790, 2016. 2 [22] Luis Silvestre. Regularity of the obstacle problem for a fractional power of the Laplace operator. Comm. Pure Appl. Math., 60(1):67–112, 2007. 2 [23] Pawel Sztonyk. Regularity of harmonic functions for anisotropic fractional Laplacians. Math. Nachr., 283(2):289–311, 2010. 2

ASYMPTOTIC MEAN VALUE PROPERTIES FOR ANISOTROPIC OPERATORS

(C. Bucur) School of Mathematics and Statistics University of Melbourne 813 Swanston Street, Parkville VIC 3010, Australia E-mail address: [email protected] (M. Squassina) Dipartimento di Matematica e Fisica ` Cattolica del Sacro Cuore Universita Via dei Musei 41, I-25121 Brescia, Italy E-mail address: [email protected]

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