Serrin’s overdetermined problem and constant mean curvature surfaces

• Published in: Duke mathematical journal Vol. 164; no. no. 14; pp. 2643 - 2722
• Main Authors: Del Pino, Manuel, Pacard, Frank, Wei, Juncheng
• Format: Journal Article
• Language: English
• Published: Duke University Press 2015
• Subjects: 35J25; 35J67; constant mean curvature surface; entire minimal graph; overdetermined elliptic equation
• ISSN: 1547-7398
• DOI: 10.1215/00127094-3146710

For all N\geq9 , we find smooth entire epigraphs in \mathbb{R}^{N} , namely, smooth domains of the form \Omega:=\{x\in\mathbb{R}^{N}|x_{N}\gt F(x_{1},\ldots,x_{N-1})\} , which are not half-spaces and in which a problem of the form \Delta u+f(u)=0 in \Omega has a positive, bounded solution with 0 Dirichlet boundary data and constant Neumann boundary data on \partial\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 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.