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Characterization of balls by Riesz-potentials. (English) Zbl 1180.31008

This paper presents a nice characterization of balls in \(\mathbb R^N\), \(N\geq 2\), in terms of generalized Riesz potentials. More precisely, let \(\Omega\subset\mathbb R^N\) be a bounded convex domain and \(u(x)=\int_\Omega |x-y|^{\alpha-N}dy\), \(0<\alpha\neq N\). It is shown in this paper that \(\Omega\) is a ball if and only if \(u\) is constant on \(\partial\Omega\). In case \(\alpha=N\) a similar result is obtained replacing \(u\) with \(u(x)=-\int_\Omega \log|x-y|dy\). These findings extend some previous results due to L.E. Fraenkel [Introduction to maximum principles and symmetry in elliptic problems, Cambridge Tracts in Mathematics, vol. 18, Cambridge: Cambridge University Press (2000; Zbl 0947.35002)] for the case \(\alpha=2\). The proofs rely on an adapted version of the moving plane method for integral representations of solutions to higher order differential equations in terms of Green functions.

MSC:

31B30 Biharmonic and polyharmonic equations and functions in higher dimensions
35J30 Higher-order elliptic equations
31B35 Connections of harmonic functions with differential equations in higher dimensions
35S99 Pseudodifferential operators and other generalizations of partial differential operators

Citations:

Zbl 0947.35002
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References:

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