Biao Ou, University of Toledo Radial symmetry of positive continuous solutions to a singular integral equation.

\underline{Abstract}: (Joint with W.-X. Chen and C.-M. Li) We use the moving plane method to prove the radial symmetry of positive continuous solutions of the singular integral equation

u(x) = ∫_{R^{n}} \frac{1}{|x − y|^{n−α}} u(y)^{\frac{n+α}{n−α}} dy

where 0 < α < n. The symmetry forces the solution to take specific form of functions. The integral equation appears in the context of the Hardy-Littlewood-Sobolev inequalities and in the conformal geometry of spheres.