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In 1971, Serrin proved that the only bounded domain for which the overdetermined problem ∆u + f (u) = 0, u > 0 in Ω u = 0 on ∂Ω ∂νu = C on ∂Ω admits a solution is the bounded ball. In 1997, Berestycki, Caffarelli and Nirenberg considered the unbounded domain case, and proposed the following conjecture: If Serrin’s problem admits a solution and Ωc is connected, then Ω is either a half space, a cylinder B × RN−k, or complement of a ball or cylinder. In this talk, I shall discuss positive and negative answers to this conjecture.In particular, when Ω is an epigraph Ω = {xN > φ(x′)}, we show that (1) BCN conjecture is always true when N = 2, (2) BCN conjecture is true when 3 ≤ N ≤ 8 if ∂u > 0 (3) BCN conjecture is false when N ≥ 9. A key observation ∂xN is the connection between this problem and a one-phase free boundary problem.