# Monotonicity and Symmetry of Solutions to Fractional Laplacian in Strips

Authors: Tao Sun; Hua Su
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, using the method of moving planes, we study the monotonicity in some directions and symmetry of the Dirichlet problem involving the fractional Laplacian−Δα/2ux=fux,x∈Ω,ux>0,x∈Ω,ux=0,x∈ℝn\Ω, in a slab-like domain Ω=ℝn−1×0,h⊂ℝn.


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## Body

## 1. Introduction

The fractional Laplacian inℝn is a nonlocal pseudo-differential operator defined by
(1)−Δα/2ux=Cn,αlimε⟶0∫ℝn\Bεxux−uzx−zn+αdz,whereCn,α is a normalisation constant and α is any real number between 0 and 2. Let
(2)Lα=u:ℝn⟶ℝ1∣∫ℝnux1+xn+αdx<∞.Then, it is easy to verify that foru∈Lα∩Cloc1,1, the integral on the right-hand side of (1) is well defined. Throughout this paper, we consider the fractional Laplacian in this setting.Due to applications in physics, chemistry, biology, probability, and finance, differential equations involving the fractional Laplacian−Δα/2 have received growing attention from the mathematical communicity in recent years (see [1–14]). There are many papers devoted to the study of qualitative properties of fractional Laplacian equations in bounded or unbounded domains, but seldom are concerned with slab-like domains. For example, in [15], the authors established the symmetry and monotonicity of positive solutions of the following problem with more general nonlinearity on a bounded domain.
(3)−Δα/2ux=fux,x∈B10,ux=0,x∈ℝn\B10,using a direct method of moving planes. For local elliptic operators, these kinds of approaches were introduced decades ago in the paper [16] and then summarized in the book [17], among which the narrow region principle and the decay at infinity have been applied extensively by many researchers to solve various problems. For more articles concerning the method of moving plans for nonlocal equations, please see [18–20] and the references therein.However, there are some papers of elliptic second-order boundary value problems concerned with features like monotonicity in some directions and symmetry for positive solutions in slab-like domains. For instance, in [21], using the “sliding method,” the authors studied monotonicity in some directions and symmetry of elliptic second-order boundary value problems of the type.
(4)Δu+fu=0,x∈Ω,ux>0,x∈Ω,ux=0,x∈∂Ω,in a slabΩ=ℝ×0,h⊂ℝ2. For more articles concerning the “sliding method,” please see [22, 23] and the references therein.Motivated by the above work, in this paper, using the direct method of moving planes, we study the monotonicity in some directions and symmetry of fractional Laplacian boundary value problems of the type.(5)−Δα/2ux=fux,x∈Ω,ux>0,x∈Ω,ux=0,x∈ℝn\Ω,in a class of special unbounded domains Ω of ℝn: infinite cylinders or more generally, product domains of the form
(6)Ω=ℝn−j×ω,where ω is a smooth bounded domain in ℝj.We denote the variables inΩ by x′,y, x′∈ℝn−j, and y∈ω⊂ℝj with j≥1. It is not assumed that Ω is bounded. The function f appearing in (5) will always be assumed to be (globally) Lipschitz continuous. We firmly believe that the result introduced here is of great importance, and the ideals and methods can be applied to study a variety of nonlocal problems with more general operators and nonlinearities.In most of what follows, we consider the casej=1. In this case, the proof of monotonicity and symmetry yields the following statement for j=1.Theorem 1.
Let(7)Σ=x′,y∣x′∈ℝn−1,0<y<h.
Supposeu∈Lα∩Cloc1,1Σ satisfies
(8)−Δα/2u=fu,inΣ,ux>0,inΣ,ux=0,inℝn\Σ,with f· being Lipschitz continuous. Then, for any positive l<h/2,
(9)ux′,y<ux′,2l−y,in∑l=x′,y∣x′∈ℝn−1,0<y<l,and u is symmetric in y about y=h/2.
If we further assume thatu∈Cloc3Σ¯h/2, then
(10)∂u∂y>0,in∑h/2=x′,y∣0<y<h2.Remark 2.
Here, the domainΩ is an infinite cylinder, and it is more general than the usual unbounded domains. For instance, if we let h⟶∞ in Theorem 1, we can get monotonicity of positive solutions of the Dirichlet problem involving the fractional Laplacian in the half space.


## 2. Preliminaries and Lemmas

LetTλ be a hyperplane in ℝn. Without loss of generality, we may assume that
(11)Tλ=x=x′,y∈ℝn−1×0,h∣y=λ,∑λ=x=x′,y∈ℝn−1×0,h∣0<y<λ.And forx′,y∈Σλ, we let xλ=x′,2λ−y be the reflection of x about the plane Tλ. Denote wλx=uxλ−ux. For simplicity of notation, in the following, we denote wλ by w and Σλ by Σ.Lemma 3 (Narrow region principle [15]).
LetΩ be a bounded narrow region in Σ, such that it is contained in x∣λ−l<y<λ with small l. Suppose that w∈Lα∩Cloc1,1Ω and is lower semicontinuous on Ω¯. If cx is bounded from below in Ω and
(12)−Δα/2wx+cxwx≥0inΩ,wx≥0inΣ\Ω,wxλ=−wxinΣ,then for sufficiently small l, we have
(13)wx≥0inΩ.
Furthermore, ifw=0 at some point in Ω, then
(14)wx=0almosteverywhereinℝn.
These conclusions hold for unbounded regionΩ if we further assume that
(15)lim¯∣x∣⟶∞wx≥0.Lemma 4 (A Hopf type lemma for antisymmetric functions [24]).
Assume thatw∈Cloc3Σ¯, lim¯x⟶∂Σcx=o1/distx,∂Σ2, and
(16)−Δα/2wx+cxwx=0inΣ,wx≥0inΣ,wxλ=−wxinΣ.
Then,(17)∂w∂ν<0,x∈∂Σ.


## 3. Proof of Theorem1

Proof of Theorem 1.
Now we carry on the method of moving planes on the solutionu along y direction.Step 1.
We show that, for sufficiently smallλ>0,
(18)wλx>0,x∈∑λ,where wλx=uxλ−ux.As usual, we can easily verify thatwλ satisfies the following linear equation
(19)−Δα/2wλ+cλxwλ=0,x∈∑λ.Indeed,uxλ satisfies the same equation in (8) as ux; thus, (19) is obtained by subtracting one from the other and letting
(20)cλx=fuxλ−fuxux−uxλ,ux≠uxλ,0,ux=uxλ.By the assumption thatf is (globally) Lipschitz continuous, with some Lipschitz constant b, we have
(21)cλL∞Σλ≤b,∀λ∈0,h2.From the narrow region principle, we can easily know that for sufficiently smallσ>0,
(22)wλx≥0,∀x∈∑λ,λ∈0,σ.Furthermore, it follows fromwλx′,0>0 that we have
(23)wλx>0,∀x∈∑λ,λ∈0,σ.Step 2.
The proof in Step1 provides a starting point, from which we can now move the plane Tλ to the right as long as (18) holds to its limiting position.Let(24)λ0=supλ∈0,h2∣wμx>0,∀x∈∑μ,μ≤λ.In this part, we show that(25)λ0=h2,wλ0x≡0,x∈∑λ0.Suppose thatλ0<h/2, we show that the plane Tλ can be moved further. To be more rigorous, we only need to prove that there exists ε>0, such that for any λ∈λ0,λ0+ε, we have
(26)wλ0x>0,x∈∑λ0.This is a contradiction with the definition ofλ0. Hence, we have λ0=h/2.Now we prove (26) by the narrow region principle (Lemma 3). By the definition of λ0, we can easily have
(27)wλ0x≥0,x∈∑λ0.In fact, whenλ0<h/2, we have
(28)x>011wλ0x>0,x∈∑λ0.If not, there existsx^ such that
(29)wλ0x^=minΣλ0wλ0x=0.Then, we have(30)−Δα/2wλ0x^=Cn,αPV∫ℝn−wλ0zx^−zn+αdz=Cn,αPV∫Σλ0−wλ0zx^−zn+αdz+∫ℝn\Σλ0−wλ0zx^−zn+αdz=Cn,αPV∫Σλ0−wλ0zx^−zn+αdz+∫Σλ0wλ0zx^−zλn+αdz=Cn,αPV∫Σλ01x^−zλn+αdz−1x^−zn+αwλ0zdz<0.On the other hand,(31)−Δα/2wλ0x^=−Δα/2ux^λ0−−Δα/2ux^=fux^λ0−fux^=0.This is a contradiction with (30). Thus, (28) holds.Then, it follows from (28) that there exists a constant c0>0 and δ>0, such that
(32)wλ0x≥c0,x∈Σ¯λ0−δ.Sincewλ depends on λ continuously, there exists ε∈0,δ, such that for all λ∈λ0,λ0+ε, we have
(33)wλx>0,x∈Σ¯λ0−δ.Then, from the narrow region principle (Lemma3), we conclude that for all λ∈λ0,λ0+ε,
(34)wλx>0,x∈Σ¯λ.This is a contradiction with the definition ofλ0. Therefore, we must have λ0=h/2, and
(35)wλ0x≡0,x∈∑λ0.Consequently, for allλ: 0<λ<h/2, we have wλ>0 in Σλ. Therefore, (9) holds, and u is symmetric in y about y=h/2.Further, if we assumeu∈Cloc3Σ¯h/2, we now prove (10) holds. Indeed, wλ satisfies the following linear equation
(36)−Δα/2wλ+cλxwλ=0,x∈∑λ,with wλx′,λ=0. Also, by the former proof, we know that wλ>0 in Σλ. Here, we consider the distance from x to the upper boundary y=λ of Σλ, denoted by distx,∂Σλ≕d. Then, dx,∂Σλ=λ−y. Thus, by (20) we know that
(37)lim¯x⟶∂Σλcxdx,∂∑λ2=lim¯x⟶∂Σλcxdλ−x22=0.Therefore,(38)lim¯x⟶∂Σλcx=o1dx,∂Σλ2.Consequently, the Hopf type lemma for antisymmetric functions (Lemma4) leads to
(39)−2∂u∂yx′,λ≡∂wλ∂yx′,λ<0,∀x′∈ℝn−1,λ∈0,h2,which implies that (10) holds. This completes the proof.



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*Source: 5354775-2021-11-10.xml*