Liouville Theorem Involving the Uniformly Nonlocal Operator
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Liouville Theorem Involving the Uniformly Nonlocal Operator Meng Qu1
· Jiayan Wu2
Received: 18 August 2020 / Revised: 5 October 2020 / Accepted: 10 October 2020 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract We prove that u is constant if u is a bounded solution of Aα u(x) = Cn,α P.V.
Rn
a(x − y)(u(x) − u(y)) dy = 0, x ∈ Rn , |x − y|n+α
where the function a : Rn → R be uniformly bounded and radial decreasing. This result can be regarded as the generalization of usual Liouville theorem. To get the proof, we establish a maximum principle involving the nonlocal operator Aα for antisymmetric functions on any half space. Keywords Uniformly elliptic nonlocal operator · A maximum principle on unbounded domains · A Liouville theorem Mathematics Subject Classification 35J60
1 Introduction For given 0 < α < 2, the uniformly elliptic nonlocal operator is defined by Aα u(x) := Cn,α P.V.
Rn
a(x − y)(u(x) − u(y)) dy, |x − y|n+α
(1.1)
Communicated by Maria Alessandra Ragusa.
B
Meng Qu [email protected] Jiayan Wu [email protected]
1
School of Mathematics and Statistics, Anhui Normal University, Wuhu 241002, China
2
School of Mathematics, Zhejiang University, Hangzhou 310000, China
123
M. Qu, J. Wu
where Cn,α is a constant depending on n and α, P.V. stands for the Cauchy principal value and the function a : Rn → R satisfies the following condition. (a1 ): (uniformly bounded) for some constants m, M > 0, we have m ≤ a(x) ≤ M < ∞, ∀x ∈ Rn , (a2 ): (radial decreasing) for any x, y ∈ Rn with |x| ≤ |y|, then a(x) ≥ a(y), where |x| = (x12 + x22 + · · · + xn2 )/1/2 . 1,1 If we assume u ∈ Cloc ∩ L α (Rn ), where L α is defined as
L α (Rn ) = u ∈ L 1loc |
Rn
|u(x)| dx < ∞ , 1 + |x|n+α
then the above integral in (1.1) is well defined. The operator Aα arises from purely jump Levy processes. It also appears in the research on stochastic control problems and stochastic games; see [5]. Caffarelli and Silvestre have proved C 1,α regularity result for the nonlocal equations involving Aα in [5,6]. If a(x) = 1, then the operator Aα becomes the traditional fractional Laplacian α α operator (−) 2 . Compared with the structure of , the nonlocal nature of (−) 2 α makes the equations involving (−) 2 difficult to study. In 2007, Caffarelli and Silvestre [4] introduced so-called the extension method which turns the nonlocal problem into a local one in higher dimensions. One can also use the integral equations method, like the method of moving planes in integral forms and regularity lifting to investigate some properties of the equation that involves the fractional Laplacian operator; see [10,13,17]. Recently, equations involving the fractional Laplacian have been extensively studied, which were used to model diverse physical phenomena, such as quasi-geostrophic flow, anomalous diffusion, pseudo-relativistic boson stars and boundary control problems; see [1,7,8,23,24,26] and references therein. However, it seems that the extension method and integral equations method
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