An overdetermined problem in Riesz-potential and fractional Laplacian

The main purpose of this paper is to address two open questions raised by W. Reichel on characterizations of balls in terms of the Riesz potential and fractional Laplacian. For a bounded $C^1$ domain $\mathrm{\Omega }\subset {\mathbb{R}}^N$, we consider the Riesz-potential $$u(x)={\int }_{\mathrm{\Omega }}\frac{1}{\mathrm{\mid }x-y{\mathrm{\mid }}^{N-\alpha }}dy$$ for $2\le \alpha ≠N$. We show that $u=$ constant on the boundary of $\mathrm{\Omega }$ if and only if $\mathrm{\Omega }$ is a ball. In the case of $\alpha =N$, the similar characterization is established for the logarithmic potential. We also prove that such a characterization holds for the logarithmic Riesz potential. This provides a characterization for the overdetermined problem of the fractional Laplacian. These results answer two open questions of W. Reichel to some extent.