Serrin’s overdetermined problem and constant mean curvature surfaces

Abstract

For all N >= 9, we find smooth entire epigraphs in R-N, namely, smooth domains of the form Omega := {x is an element of R-N broken vertical bar X-N broken vertical bar > F(X-1, . . . , X-N-1)}, which are not half-spaces and in which a problem of the form Au f (u) = 0 in 2 has a positive, bounded solution with 0 Dirichlet boundary data and constant Neumann boundary data on partial derivative Omega. This answers negatively for large dimensions a question by Berestycki, Caffarelli, and Nirenberg. In 1971, Serrin proved that a bounded domain where such an overdetern2ined 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 (CMG) 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 d given CMC surface where Serrin's overdetermined problem is solvable.