Abstract
For a bounded convex domain \({G\subset\mathbb{R}^N}\) and \({2 < \alpha\not = N}\) consider the unit- density Riesz-potential \({u(x) = \int_G|x-y|^{\alpha-N}\,dy}\) . We show in this paper that u = const. on ∂G if and only if G is a ball. This result corresponds to a theorem of L.E. Fraenkel, where the ball is characterized by the Newtonian-potential (α = 2) of unit density being constant on ∂G. In the case α = N the kernel |x − y|α-N is replaced by − log|x − y| and a similar characterization of balls is given. The proof relies on a recent variant of the moving plane method which is suitable for Green-function representations of solutions of (pseudo-)differential equations of higher-order.
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Reichel, W. Characterization of balls by Riesz-potentials. Annali di Matematica 188, 235–245 (2009). https://doi.org/10.1007/s10231-008-0073-6
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DOI: https://doi.org/10.1007/s10231-008-0073-6