On Serrin’s Overdetermined Problem and a Conjecture of Berestycki, Caffarelli and Nirenberg

This paper concerns rigidity results to Serrin’s overdetermined problem in an epigraph  ∆u + f(u) = 0, in Ω = {(x′, xn) : xn > φ(x′)}, u > 0, in Ω, u = 0, on ∂Ω, |∇u| = const., on ∂Ω. We prove that up to isometry the epigraph must be an half space and that the solution u must be one-dimensional, provided that one of the following assumptions are satisfied: either n = 2; or φ is globally Lipschitz, or n ≤ 8 and ∂u ∂xn > 0 in Ω. In view of the counterexample constructed in [9] in dimensions n ≥ 9 this result is optimal. This partially answers a conjecture of Berestycki, Caffarelli and Nirenberg [5].