## 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

### Keywords:

Riesz potentials; moving plane method; radial symmetry

Zbl 0947.35002
Full Text:

### References:

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