OVERDETERMINED PROBLEMS WITH FRACTIONAL LAPLACIAN

Let N >= 1 and s is an element of (0, 1). In the present work we characterize bounded open sets Omega with C-2 boundary (not necessarily connected) for which the following overdetermined problem {(-Delta)(s) u = f(u) in Omega; u = 0 in R-N \ Omega; (partial derivative(eta))(s)u = Const. on partial derivative Omega has a nonnegative and nontrivial solution, where. is the outer unit normal vectorfield along partial derivative Omega and for x(0) is an element of partial derivative Omega (partial derivative(eta))(s) u(x(0)) = - lim(t -> 0) u(x(0) - t(eta)(x(0)))/t(s). Under mild assumptions on f, we prove that Omega must be a ball. In the special case f equivalent to 1, we obtain an extension of Serrin's result in 1971. The fact that Omega is not assumed to be connected is related to the nonlocal property of the fractional Laplacian. The main ingredients in our proof are maximum principles and the method of moving planes.