Entire \(s\)-harmonic functions are affine (Q2796728)
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scientific article; zbMATH DE number 6560783
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Entire \(s\)-harmonic functions are affine |
scientific article; zbMATH DE number 6560783 |
Statements
29 March 2016
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fractional Laplacian
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Liouville theorem
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uniqueness
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Riesz kernel
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entire \(\alpha\)-harmonic
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Cauchy estimates
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Entire \(s\)-harmonic functions are affine (English)
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The author considers the equation NEWLINE\[NEWLINE (-\Delta)^s u=P \quad \text{in}\; {\mathcal D'}(\mathbb{R}^N),\leqno{(1)} NEWLINE\]NEWLINE where \(s\in (0,1),\;(-\Delta)^s \) is the fractional Laplacian, \(P\) is a polynomial and \({\mathcal D'}(\mathbb{R}^N)\) is the dual of \({C}_c^\infty(\mathbb{R}^N)\). The author proves that solutions of problem \((1)\) are affine and \(P=0\). Further, he proves the uniqueness of the Riesz potential \(|x|^{2s-N}\) in Lebesgue spaces.
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