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 C1C1 domain Ω⊂RNΩ⊂RN, we consider the Riesz-potential u(x)=∫Ω1∣x−y∣N−αdy for 2≤α≠N2≤α≠N. We show that u=u= constant on ∂Ω∂Ω if and only if ΩΩ is a ball. In the case of α=Nα=N, the similar characterization is established for the logarithmic potential u(x)=∫Ωlog1∣x−y∣dy. We also prove that such a characterization holds for the logarithmic Riesz potential u(x)=∫Ω∣x−y∣α−Nlog1∣x−y∣dy when the diameter of the domain ΩΩ is less than e1N−α in the case when α−Nα−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.