Serrin's overdetermined problem and constant mean curvature surfaces

For all N ≥ 9, we find smooth entire epigraphs in R^N, namely, smooth domains of the form Ω:= {x ∈ R^N |x_N >F(x₁, ..., x_N-1)}, which are not half-spaces and in which a problem of the form Δu + f (u) = 0 in Ω has a positive, bounded solution with 0 Dirichlet boundary data and constant Neumann boundary data on ∂Ω. This answers negatively for large dimensions a question by Berestycki, Caffarelli, and Nirenberg. In 1971, Serrin proved that a bounded domain where such an overdetermined problem is solvable must be a ball, in analogy to a famous result by Alexandrov that states that an embedded compact surface with constant mean curvature (CMC) in Euclidean space must be a sphere. In lower dimensions we succeed in providing examples for domains whose boundary is close to large dilations of a given CMC surface where Serrin's overdetermined problem is solvable.