An overdetermined problem in Riesz-potential and fractional Laplacian

The main purpose of this paper is to address two open questions raised by Reichel (2009) in [2] on characterizations of balls in terms of the Riesz potential and fractional Laplacian. For a bounded C1 domainΩ ⊂ RN , we consider the Riesz-potential u(x) = ∫ Ω 1 | x− y |N−α dy for 2 ≤ α = N . We show that u = constant on ∂Ω if and only if Ω is a ball. In the case of α = N , the similar characterization is established for the logarithmic potential u(x) = ∫ Ω log 1 |x−y| dy. We also prove that such a characterization holds for the logarithmic Riesz potential u(x) = ∫ Ω | x− y |α−N log 1 | x− y | dy when the diameter of the domain Ω is less than e 1 N−α in the case when α − N is a nonnegative even integer. This provides a characterization for the overdetermined problem of the fractional Laplacian. These results answer two open questions in Reichel (2009) [2] to some extent. Moreover, we also establish some nonexistence result of positive solutions to a class of integral equations in an exterior domain. © 2011 Elsevier Ltd. All rights reserved.