Monotonicity and Symmetry of Solutions to Fractional Laplacian in Strips

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 in a slab-like domain .

1. Introduction

The fractional Laplacian in is a nonlocal pseudo-differential operator defined by

where is a normalisation constant and is any real number between 0 and 2. Let

Then, it is easy to verify that for , 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 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. 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.

in a slab . 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. in a class of special unbounded domains of : infinite cylinders or more generally, product domains of the form where is a smooth bounded domain in .

We denote the variables in by , , and with . It is not assumed that is bounded. The function 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 case . In this case, the proof of monotonicity and symmetry yields the following statement for .

Theorem 1. Let Suppose satisfies with being Lipschitz continuous. Then, for any positive , and is symmetric in about .
If we further assume that , then

Remark 2. Here, the domain is an infinite cylinder, and it is more general than the usual unbounded domains. For instance, if we let 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

Let be a hyperplane in . Without loss of generality, we may assume that

And for , we let be the reflection of about the plane . Denote . For simplicity of notation, in the following, we denote by and by .

Lemma 3 (Narrow region principle [15]). Let be a bounded narrow region in , such that it is contained in with small l. Suppose that and is lower semicontinuous on . If is bounded from below in and then for sufficiently small l, we have Furthermore, if at some point in , then These conclusions hold for unbounded region if we further assume that

Lemma 4 (A Hopf type lemma for antisymmetric functions [24]). Assume that , , and Then,

3. Proof of Theorem 1

Proof of Theorem 1. Now we carry on the method of moving planes on the solution along direction.

Step 1. We show that, for sufficiently small , where .

As usual, we can easily verify that satisfies the following linear equation

Indeed, satisfies the same equation in (8) as ; thus, (19) is obtained by subtracting one from the other and letting

By the assumption that is (globally) Lipschitz continuous, with some Lipschitz constant , we have

From the narrow region principle, we can easily know that for sufficiently small ,

Furthermore, it follows from that we have

Step 2. The proof in Step 1 provides a starting point, from which we can now move the plane to the right as long as (18) holds to its limiting position.

Let

In this part, we show that

Suppose that , we show that the plane can be moved further. To be more rigorous, we only need to prove that there exists , such that for any , we have

This is a contradiction with the definition of . Hence, we have .

Now we prove (26) by the narrow region principle (Lemma 3). By the definition of , we can easily have

In fact, when , we have

If not, there exists such that

Then, we have

On the other hand,

This is a contradiction with (30). Thus, (28) holds.

Then, it follows from (28) that there exists a constant and , such that

Since depends on continuously, there exists , such that for all , we have

Then, from the narrow region principle (Lemma 3), we conclude that for all ,

This is a contradiction with the definition of . Therefore, we must have , and

Consequently, for all : , we have in . Therefore, (9) holds, and is symmetric in about .

Further, if we assume , we now prove (10) holds. Indeed, satisfies the following linear equation with . Also, by the former proof, we know that in . Here, we consider the distance from to the upper boundary of , denoted by . Then, . Thus, by (20) we know that

Therefore,

Consequently, the Hopf type lemma for antisymmetric functions (Lemma 4) leads to which implies that (10) holds. This completes the proof.

Data Availability

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

Conflicts of Interest

The authors declare that there is no conflict of interest regarding the publication of this paper.

Authors’ Contributions

All authors contributed equally to the manuscript and typed, read, and approved the final manuscript.

Acknowledgments

The author was supported by the Project of National Science Foundation of China and the Project of Shandong Province Higher Educational Science and Technology Program.