A Rigidity Result for Overdetermined Elliptic Problems in the Plane

Let f:[0,+) be a (locally) Lipschitz function and Omega subset of R-2 a C-1,C-alpha domain whose boundary is unbounded and connected. If there exists a positive bounded solution to the overdetermined elliptic problem {Delta u + f(u) = 0 in Omega, u = 0 on partial derivative Omega, partial derivative u/partial derivative(nu) over bar = 1 on partial derivative Omega, we prove that Omega is a half-plane. In particular, we obtain a partial answer to a question raised by H. Berestycki, L. Caffarelli, and L. Nirenberg in 1997.(c) 2017 Wiley Periodicals, Inc.