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A Liouville Theorem for the Fractional Laplacian

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Author(s): Ran Zhuo , Wenxiong Chen , Xuewei Cui , Zixia Yuan

Publication date Created: 28 January 2014

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Abstract

We extend the classical Liouville Theorem from Laplacian to the fractional Laplacian, that is, we prove Every \(\alpha\)-harmonic function bounded either above or below in all of \(R^n\) must be constant.

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Most cited references 7

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An extension problem related to the fractional Laplacian

Luis A. Caffarelli, Luís Silvestre (2006)

The operator square root of the Laplacian \((-\lap)^{1/2}\) can be obtained from the harmonic extension problem to the upper half space as the operator that maps the Dirichlet boundary condition to the Neumann condition. In this paper we obtain similar characterizations for general fractional powers of the Laplacian and other integro-differential operators. From those characterizations we derive some properties of these integro-differential equations from purely local arguments in the extension problems.

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Classification of solutions for an integral equation

Wenxiong Chen, Congming Li, Biao Ou (2006)

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The principal eigenvalue and maximum principle for second-order elliptic operators in general domains

H. Berestycki, L. Nirenberg, S. Varadhan (1994)

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Author and article information

Journal

Publication date Created: 28 January 2014

Article

ArXiV ID: 1401.7402

SO-VID: 987e1c3e-44fa-42ee-9a04-4ac9c3fa819c

License:

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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ACM classes 35J61, 35S05, 45K05

Comments 14 pages

Categories math.AP

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