Marcinkiewicz estimates for solution to fractional elliptic Laplacian equation

Computers and Mathematics with Applications 78 (2019) 1732–1738

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Marcinkiewicz estimates for solution to fractional elliptic Laplacian equation✩ ∗

Shuibo Huang , Qiaoyu Tian School of Mathematics and Computer, Northwest Minzu University, Lanzhou, Gansu 730000, PR China

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a b s t r a c t In this paper, we consider the Marcinkiewicz summability of solutions to the following fractional elliptic problem

Article history: Available online 16 May 2019

⎧ ⎨ (−∆)s u = f (x), u > 0, ⎩ u = 0,

Keywords: Fractional Laplacian Marcinkiewicz estimates

∈ Ω, ∈ Ω, x ∈ RN \ Ω ,

where (−∆)s denotes the fractional Laplacian operator, s ∈ (0, 1), Ω ⊂ RN is a bounded domain with Lipschitz boundary, f belongs to some Marcinkiewicz space Mm (Ω ) with m > 1. The main novelty of this paper is actually the fact that the solutions to the above )2 ( N , instead of m > 2s . The results of this paper are equation are bounded if m > N2N +2s new even for s = 1. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction The main goal of this work is to study the Marcinkiewicz summability of solutions to the following fractional elliptic problem

⎧ ⎨ (−∆)s u = f (x), u > 0, ⎩ u = 0,

x ∈ Ω, x ∈ Ω, x ∈ RN \ Ω ,

(1.1)

where Ω ⊂ RN is a bounded domain with Lipschitz boundary, f ∈ Mm (Ω ) with m > 1, here Mm (Ω ) denotes the Marcinkiewicz space, (−∆)s is fractional Laplacian operator, s ∈ (0, 1), defined as (−∆)s u = aN ,s P .V .

∫

u(x) − u(y) RN

|x − y|N +2s

,

where P.V. stands for the Cauchy principal value, aN ,s is a normalization constant. Recently, regularity of solutions to fractional elliptic problem has been extensively studied, see for example [1–5] and references therein. Recall that, if f belongs to the Lebesgue space Lm (Ω ) with m ≥ 1, summability results of solutions u to problem (1.1) are proved in [6,7].For corresponding results of classical Laplace operator, that is s = 1, see [8,9] and the ✩ This research was partially supported by the National Science Foundation of China (No. 11761059), Fundamental Research Funds for the Central Universities. (No.31920170001) ∗ Corresponding author. E-mail addresses: [email protected] (S. Huang), [email protected] (Q. Tian). https://doi.org/10.1016/j.camwa.2019.04.032 0898-1221/© 2019 Elsevier Ltd. All rights reserved.

S. Huang and Q. Tian / Computers and Mathematics with Applications 78 (2019) 1732–1738

references therein. The aim of this paper is to f ∈ Mm (Ω ) with m > 1. It is worthwhile to interest in the Marcinkiewicz regularity based most naturally regarded in the Marcinkiewicz The main results are

1733

study the Marcinkiewicz regularity of solutions to problem (1.1) provided point out that Lm (Ω ) ⊂ Mm (Ω ) ⊂ Ln (Ω ) for any 1 ≤ n < m < ∞. The on the fact that, the fundamental solution of fractional Poisson equation is space, rather than Lebesgue space.

Theorem 1.1. Suppose that Ω ⊂ RN is a bounded domain with Lipschitz boundary and f belongs to the Marcinkiewicz space Mm (Ω ).

)2

(1) If m > N2N . Then there exists a constant C , only depending on N , Ω , s, ∥u∥H s (Ω ) , ∥f ∥Mm (Ω ) , such that the energy +2s 0 solution to (1.1) satisfies

(

∥u∥L∞ (Ω ) ≤ C .

(1.2)

)2

(2) If (2∗s )′ < m < N2N . Then there exists a constant C , only depending on N , Ω , s, ∥u∥H s (Ω ) , ∥f ∥Mm (Ω ) , such that the +2s 0 energy solution to (1.1) satisfies

(

∥u∥Mm∗∗ (Ω ) ≤ C ∥f ∥Mm (Ω ) , where 2∗s = 1 m∗∗

2N N −2s

=

1 2∗

(1.3)

is the so called fractional critical Sobolev exponent, m∗∗ satisfies

−

m2∗s − m − 2∗s

s

m(2∗s − 1)

.

(3) If 1 < m < (2∗s )′ . Then there exists a constant C , only depending on N , Ω , s, ∥f ∥Mm (Ω ) , such that the weak solution to (1.1) satisfies (1.3) and

∥(−∆)s/2 u∥Mm∗ (Ω ) ≤ C ∥f ∥Mm (Ω ) ,

(1.4)

where m∗ =

4mN(N + 2s) 4N 2

+ (N + 2s)[2N − m(N + 2s)]

.

(1.5)

Remark 1.2. The surprising character of Theorem 1.1 lies in the fact that, the boundedness of energy solutions was obtained in [6] through the Moser and Stampacchia methods if f ∈ Lm (Ω ) with m > N /2s, while in this paper, we ( )2 show the boundedness of energy solutions to problem (1.1) if f ∈ Mm (Ω ) with m > N2N . It can be easily seen that +2s

(

2 2N N +2s

)

<

N 2s

since 2s < N. mN

Furthermore, when f ∈ Lm (Ω ) with (2∗s )′ < m < N /2s, according to Theorem 16 in [6], we know that u ∈ L N −2ms (Ω ). ∗∗ mN if m > (2∗s )′ . Under the assumptions of Theorem 1.1, we show that u ∈ Mm (Ω ). Obviously, m∗∗ < N − 2ms Remark 1.3. The framework of Stampacchia [10] shows that if f ∈ Mm (Ω ) with u to

{

div(M(x)∇ u) = f (x), u = 0,

x ∈ Ω, x ∈ ∂Ω,

2N N +2


N , 2

then the weak solutions

(1.6)

mN

belong to M N −2m (Ω ), and u ∈ L∞ (Ω ) if f ∈ Mm (Ω ) with m > N2 , where M(x) is a bounded, elliptic matrix. An extension of this result to f ∈ Mm (Ω ) with 1 < m < N2N was given by Boccardo [8]. With a little modification, we can show that +2 the main results of Theorem 1.1 also hold for (1.6). More results of classical Laplace operator −∆, see [8,9]. Remark 1.4. It is interesting to note that, we prove (1.2) through the Stampacchia methods. This method has been applied successfully to study elliptic equations, involving the fractional Laplacian, and a series of fruitful results have been obtained, see [6,11,12] and the references therein. ∗ ′ It can be easily seen that Mm (Ω ) ⊂ L(2s ) (Ω ) ⊂ H −s (Ω ) provided (2∗s )′ < m. Therefore, we consider the finite energy solution in this case. For 1 < m < (2∗s )′ , we cannot expect the solution with finite energy. Thus we consider weak solution to (1.1) in this case. Remark 1.5. According to Theorem 23 of [6], we know that problem (1.1) has a unique weak solution provided f ∈ L1 (Ω ). Unfortunately, no corresponding Marcinkiewicz estimate is available if f ∈ M1 (Ω ).

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Remark 1.6. The main results of the above theorem also hold for integro-differential operators with general kernel L(u)(x) = P .V .

∫ Ω

(u(x) − u(y))K (x, y)dy,

where K (x, y) : RN × RN \ {(x, x) : x ∈ RN } → [0, +∞) satisfies (i) K (x, y) = K (y, x). (ii) There exist 0 < s < 1 and 0 < λ ≤ 1 such that for any (x, y) ∈ RN × RN , x ̸ = y, λ ≤ K (x, y)|x − y|N +2s ≤ λ−1 . Remark 1.7. By a straightforward modification of the arguments of the proof of Theorem 1.1, and taking into account ground state representation and the fractional Hardy inequality, we can obtain similar results of (1.3)–(1.4) to the following nonlocal elliptic problem with Hardy potential

⎧ u s ⎪ ⎨ (−∆) u − λ 2s = f (x), |x| u > 0, ⎪ ⎩ u = 0,

x ∈ Ω, (1.7)

x ∈ Ω, x ∈ RN \ Ω .

We must emphasize that the Hardy potential has important effect on the boundedness of solution to (1.7). In fact, the solution to (1.7) is singular near origin provided f ∈ Lm (Ω ) with m > N /2s. For more details see [11,12].For some other results on nonlocal elliptic problem with Hardy potential see [13–17]. Remark 1.8. According to the definition of fractional Laplacian operator and problem (1.1), we know that (1.3) can be regarded as global estimate since u = 0 for any x ∈ RN \ Ω , while for (1.4), we cannot get any summability information of (−∆)s/2 u outside Ω due to the nonlocal nature of fractional Laplacian operator. The paper is organized as follows. In Section 2, we present some definitions and preliminary tools, which will be used in the proof of Theorem 1.1. The proof of Theorem 1.1 is given in Section 3. 2. Useful tools and function setting In the whole paper, the letter C will denote a positive constant, not necessarily the same everywhere. According to the definition of the fractional Laplacian, it is natural to consider the following Sobolev space. For any p ∈ [1, ∞) and Ω ⊂ RN , we define W s,p (Ω ) as follows,

{ W

s,p

(Ω ) =

|u(x) − u(y)|

u ∈ L (Ω ) : p

N

|x − y| p +s

} ∈ L (Ω × Ω ) . p

When p = 2, the fractional Sobolev spaces W s,2 (Ω ) = H s (Ω ) turn out to be the Hilbert spaces. We define now the space H0s (Ω ) as the completion of C0∞ (Ω ) with respect to the norm of H s (Ω ). Notice that if u ∈ H0s (Ω ), we have u = 0 a.e. in RN \ Ω and we can write

∫ RN ×RN

|u(x) − u(y)|2 = |x − y|N +2s

∫ Q

|u(x) − u(y)|2 , |x − y|N +2s

where Q := RN × RN \ (Ω c × Ω c ). We also use the relation between the norm in the space H0s (Ω ) and the L2 -norm of the fractional Laplacian, s

1 2 2 ∥u∥2H s (Ω ) = 2a− N ,s ∥(−∆) u∥L2 (Ω ) . 0

Moreover, we can define the associated scalar product as the natural duality product between H0s and H −s :

⟨(−∆)s u, w⟩H s (RN ) := P .V .

∫

0

(u(x) − u(y))(w (x) − w (y))

∫ RN

RN

|x − y|N +2s

dxdy,

(2.1)

where H −s (RN ) is the dual space of H0s (RN ). We briefly recall some important results on Marcinkiewicz spaces. Definition 2.1. Marcinkiewicz space Mm (Ω ), m > 0, is the space of measurable functions v on Ω , such that, there exists C > 0, for any t > 0,

|{x ∈ Ω : |v (x)|≥ t }| ≤

c tm

,

endowed with the norm ∥v∥Mm (Ω ) = sup λµ{|v| > λ}1/m .

S. Huang and Q. Tian / Computers and Mathematics with Applications 78 (2019) 1732–1738

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Obviously, for 1 ≤ n < m < ∞, Lm (Ω ) ⊂ Mm (Ω ) ⊂ Ln (Ω ). For 0 < q < p < ∞ and Ω is subset of RN with µ(Ω ) < ∞, we have. q p ∥f ∥Lq (Ω ) ≤ µ(Ω )1− p ∥f ∥Mp (Ω ) (2.2) p−q For f ∈ H −s (Ω ), we say that u ∈ H0s (Ω ) is a finite energy solution to (1.1) if

Definition 2.2. aN , s 2

⟨(−∆)s u, v⟩ = ⟨f , v⟩,

∀v ∈ H0s (Ω ),

(2.3)

where ⟨·, ·⟩ represents the duality between H0s (Ω ) and H −s (Ω ), defined by (2.1). Denote T := {φ : RN → R|(−∆)s φ = ϕ, ϕ ∈ L∞ ∩ C α (Ω ), α ∈ (0, 1), φ = 0 in RN \ Ω }.

Definition 2.3. Assume f ∈ L1 (Ω ). We say that u ∈ L1 (Ω ) is a weak solution of problem (1.1) if for all nonnegative φ ∈ T , the following inequality holds,

∫ Ω

∫

u(−∆)s φ dx =

f φ dx.

Ω

The following results will be used in the proof of the main theorem. Let s1 , s2 ≥ 0 and a ≥ 0. Then

Lemma 2.4.

4a

(s1 − s2 )(sa1 − sa2 ) ≥

(a +

1)2

a+1

a+1

(s1 2 − s2 2 )2 .

(2.4)

Proof. The complete proof can be seen in [12,18,19]. □ Now, we recall the Sobolev embedding theorem. For the complete proof of the following lemma see [20,21]. Let s ∈ (0, 1) and N > 2s. There exists a constant S = S (N , s) such that for any function u ∈ H0s (Ω ) we have

Lemma 2.5.

∥u∥22∗s L

∫ ∫ (Ω )

≤S Ω

Ω

|u(x) − u(y)|2 dxdy. |x − y|N +2s

(2.5)

For any k ≥ 0, define Tk (σ ) = max{−k; min{k, σ }} and Gk (σ ) = σ − Tk (σ ). Lemma 2.6 (Lemma 4 in [6]). Let v (x) be a positive measurable function in RN . Then for any Ω ⊂ RN ,

(Tk (v (x)) − Tk (v (y))) (Gk (v (x)) − Gk (v (y))) ≥ 0.

(2.6)

Lemma 2.7 (Lemma 4.1 in [10]). Let ψ : R+ → R+ be a nonincreasing function such that

ψ (h) ≤

M ψ δ (k) (h − k)γ

, δγ

for any h > k > 0, where M > 0, δ > 1 and γ > 0. Then ψ (d) = 0, where dγ = M ψ δ−1 (0)2 δ−1 . Define Ak (u) := {x ∈ Ω : |u(x)| > k}. We have the following technical result. Lemma 2.8 (Lemma 7.2 in [22]). Let u ∈ L1 (Ω ) and set g(k) = ∥Gk (u)∥L1 . Then g is a.e. differentiable and g ′ (k) = −µ(Ak ). 3. Proof of main theorem In this section, we give the proof of Theorem 1.1. We start by the following existence results. Lemma 3.1. Let Ω ⊂ RN is a bounded domain with Lipschitz boundary and f belongs to the Marcinkiewicz space Mm (Ω ) with m > (2∗s )′ . Then problem (1.1) has a unique finite energy solution u ∈ H0s (Ω ). Proof. The proof is standard, for the sake of completeness we include the proof. According to Remark 1.4, we know that Mm (Ω ) ⊂ H −s (Ω ) if m > (2∗s )′ . Thus define bilinear form

Ψ (u, w ) =

aN , s 2

∫

(u(x) − u(y))(w (x) − w (y))

∫ RN

RN

|x − y|N +2s

dxdy.

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S. Huang and Q. Tian / Computers and Mathematics with Applications 78 (2019) 1732–1738

It is not hard to see that Ψ (u, w ) is symmetric, continuous and coercive. Thus the classical Lax–Milgram theorem implies that, for any f ∈ H −s (Ω ), there exists a unique u ∈ H0s (Ω ) such that

∫

aN , s 2

(u(x) − u(y))(w (x) − w (y))

∫

RN

|x − y|N +2s

RN

dxdy = ⟨f , w⟩,

∀w ∈ H0s (Ω ).

This fact, together with Definition 2.2, leads to the existence of problem (1.1). □ Now we establish L∞ estimates of solution to problem (1.1) by Stampacchia methods. If f ∈ Lm (Ω ) with m > N /2s, the following lemma was obtained in [6] via two different method: the Moser method and the Stampacchia method. Lemma 3.2. Suppose that Ω ⊂ RN be an arbitrary bounded open set and f belongs to the Marcinkiewicz space Mm (Ω ) with m>

(

)2

2N N + 2s

.

(3.1)

Then there exists a constant C , only depending on N , Ω , s, ∥u∥H s (Ω ) , ∥f ∥Mm (Ω ) , such that the energy solution to problem (1.1) 0 satisfies

∥u∥L∞ (Ω ) ≤ C .

(3.2)

Proof. The existence of solution follows by Lemma 3.1. Furthermore, u ∈ H0s (Ω ). Thus, for any k > 0, we can choose w = Gk (u) as a test function in the energy formulation of (2.3). Thus, Sobolev inequality, together with (2.6) and the Hölder inequality, implies that S

−2

∫

2

∥Gk (u)∥ 2∗s L

(Ω )

≤

∫Q

|Gk u(x) − Gk u(y)|2 dxdy |x − y|N +2s f (x)Gk (u)dx

≤ Ak

≤∥f ∥Mm (Ak ) ∥Gk (u)∥L2∗s (A ) ∥1∥Lp1 ,p2 (Ak ) k

≤

m(2∗s − 1)

∥f ∥Mm (Ω ) ∥Gk (u)∥L2∗s (Ω ) |Ak | ∗

m2∗s − m − 2s

∗ m2∗ s −m−2s m(2∗ s −1)

,

where Lp1 ,p2 (Ω ) is Lorentz space with indices p1 and p2 , Ak (u) := {x ∈ Ω : |u(x)| > k} and 1=

1

+

m

1 2∗s

+

1 p1

, 1=

1 2∗s

+

1 p2

.

Consequently

∥Gk (u)∥L2∗s (Ω ) ≤ S 2

m(2∗s − 1)

∥f ∥Mm (Ω ) |Ak | ∗

m2∗s − m − 2s

∗ m2∗ s −m−2s m(2∗ s −1)

.

(3.3)

For any h > k, we have that Ah ⊂ Ak and Gk (s)χAh ≥ (h − k). Therefore, 1

∗

(h − k)|Ah | 2s ≤ S 2

m(2∗s − 1)

∥f ∥Mm (Ω ) |Ak | ∗

m2∗s − m − 2s

∗ m2∗ s −m−2s m(2∗ s −1)

,

which implies that ∗

(

|Ah | ≤ S 2

m(2s − 1) m2∗s − m − 2∗s

)2∗s

2∗

s ∥f ∥M m (Ω ) |Ak |

∗ ∗ 2∗ s (m2s −m−2s ) m(2∗ s −1) ∗

(h − k)2s

.

In view of (3.1), we find 2∗s (m2∗s − m − 2∗s ) m(2∗s − 1)

> 1.

According to Lemma 2.7, we derive that there exists k0 such that ϕ (k) = |Ah | = 0 for any k > k0 , which leads to (3.2).

□

Lemma 3.3. Suppose that Ω ⊂ RN is a bounded domain with Lipschitz boundary and f belongs to the Marcinkiewicz space Mm (Ω ) with 1
(

2N N + 2s

)2

.

(3.4)

S. Huang and Q. Tian / Computers and Mathematics with Applications 78 (2019) 1732–1738

1737

Then there exists a constant C , only depending on N , Ω , s, ∥f ∥Mm (Ω ) , such that solution to (1.1) satisfies

∥u∥Mm∗∗ (Ω ) ≤ C ∥f ∥Mm (Ω ) .

(3.5)

Proof. The existence of solution was obtained in Lemma 3.1. Hence we have just to show (3.5) holds. Use Gk (u) as a test function in (2.3), we also have (3.3) holds. Define g(k) = ∥Gk (u)∥L1 . Thus, by (3.3), we find

∥Gk (u)∥L1 ≤ ∥Gk (u)∥L2∗s (Ω ) |Ak |

1− 1∗ 2 s

m2∗ −m−2∗ s 1− 1∗ + s ∗ 2 m(2 −1)

≤ C ∥f ∥Mm (Ω ) |Ak |

s

s

.

According to Lemma 2.8, we have g(k) ≤ C [−g ′ (k)]

1− 1∗∗ m

,

(3.6)

where 1 m∗∗

1

=

m2∗s − m − 2∗s

−

2∗s

m(2∗s − 1)

.

If g(k0 ) = 0 for some k0 < ∞ means that u is bounded in L∞ (Ω ). Otherwise, by (3.6), we get ∗∗

− mm ∗∗ −1

1 ≤ −C [g(k)]

g ′ (k).

Integrating in [0, k], yields k

∫

m∗∗

[g(s)]− m∗∗ −1 g ′ (s)ds,

k ≤ −C 0

which leads to

[

∗∗ 1− m m∗∗ −1

k ≤ −C g(k)

m∗∗

− g(0)1− m∗∗ −1

]

m∗∗

≤ Cg(k)1− m∗∗ −1 .

Therefore g(k) ≤

C ∗∗ −1

km

.

Thus

∫ k|A(2k)| ≤

Gk (u)dx ≤ g(k) ≤ A(2k)

C ∗∗ km −1

,

which implies that

|A(k)| ≤

C ∗∗

km

.

(3.7)

Thus by the definition of Marcinkiewicz space, we derive that (3.5) hods.

□

Lemma 3.4. Suppose that Ω ⊂ RN is a bounded domain with Lipschitz boundary and f belongs to the Marcinkiewicz space Mm (Ω ) with 1 < m < (2∗s )′ .

(3.8)

Then there exists a constant C , only depending on N , Ω , s, ∥f ∥Mm (Ω ) , such that the weak solution to (1.1) satisfies

∥(−∆)s/2 u∥Mm∗ (Ω ) ≤ C ∥f ∥Mm (Ω ) .

(3.9)

Proof. The existence of a weak solution to (1.1) is a consequence of Theorem 23 in [6] since Mm (Ω ) ⊂ L1 (Ω ) for m > 1. In the following, we only prove the estimate (3.9). Choose T1 (Gk (u)), k ≥ 1, as test function in Definition 2.3, and take into account (3.7), we have 2 aN , s

∥(−∆)s/2 u∥2L2 (B(k)) ≤

∫

1

fdx ≤ ∥f ∥Mm (Ω ) |A(k)|1− m ≤ C A(k)

∥f ∥Mm (Ω ) ∗∗ (1− 1 ) m

km

,

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S. Huang and Q. Tian / Computers and Mathematics with Applications 78 (2019) 1732–1738

where B(k) = {x ∈ Ω : k ≤ |u| ≤ k + 1} and 0 < m∗∗ 1 −

(

∥(−∆)s/2 Tk u∥2L2 (Ω ) =

k−1 ∑

1 m

)

< 1 provided (3.8) holds. Thus we derive that

∥(−∆)s/2 Tk u∥2L2 (B(i))

i=0

≤ ∥f ∥Mm (Ω ) + C

k−1 ∑ ∥f ∥Mm (Ω ) i=1

≤ Ck1−m

∗∗ (1− 1 ) m

∗∗ (1− 1 ) m

km

.

(3.10)

For fixed ρ > 0,

{x ∈ Ω : |(−∆)s/2 u| ≥ ρ} ={x ∈ Ω : |(−∆)s/2 u| ≥ ρ, |u| ≤ k} ∪ {x ∈ Ω : |(−∆)s/2 u| ≥ ρ, |u| ≤ k} ⊂{x ∈ Ω : |(−∆)s/2 Tk u| ≥ ρ, } ∪ A(k). This fact, together with (3.7) and (3.10), shows that,

|{x ∈ Ω : |(−∆)s/2 u|≥ ρ}| ∫ 1 ≤ 2 (−∆)s/2 Tk u|2 dx + |A(k)| ρ RN ∗∗ (1− 1 ) m

≤C ( ≤

k1−m

C

ρ m∗

ρ2

+

1 ∗∗

km

)

,

where m∗ is given by (1.5). This fact implies that (3.9) holds. □ Proof of Theorem 1.1. Theorem 1.1 follows directly from Lemmas 3.1–3.4. □ References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]

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