A Rigidity Result for Overdetermined Elliptic Problems in the Plane. Issue 7 (22nd April 2017) - Ros, Antonio; Ruiz, David; Sicbaldi, Pieralberto
Abstract : Let f : [ 0, + ∞ ) → ℝ be a (locally) Lipschitz function and Ω ⊂ ℝ 2 a C 1, α domain whose boundary is unbounded and connected. If there exists a positive bounded solution to the overdetermined elliptic problem [ { Δ u + f ( u ) = 0 in  Ω, u = 0 on       Ω, ∂ u ∂ ν → = 1 on   Ω, we prove that Ω 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.© 2017 Wiley Periodicals, Inc.