For all , we find smooth entire epigraphs in , namely, smooth domains of the form , which are not half-spaces and in which a problem of the form in has a positive, bounded solution with Dirichlet boundary data and constant Neumann boundary data on . 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.
Manuel Del Pino. Frank Pacard. Juncheng Wei. "Serrin’s overdetermined problem and constant mean curvature surfaces." Duke Math. J. 164 (14) 2643 - 2722, 1 November 2015. https://doi.org/10.1215/00127094-3146710