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.


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


Zbl 0947.35002
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