Serrin’s Overdetermined Problem and Constant Mean Curvature Surfaces

For all N ≥ 9, we find smooth entire epigraphs in R , namely smooth domains of the form Ω := {x ∈ R / xN > F (x1, . . . , xN−1)}, which are not half-spaces and in which a problem of the form ∆u+ 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 [3]. In 1971, Serrin [25] 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.