Serrin's overdetermined problem and constant mean curvature surfaces

Manuel Del Pino, Frank Pacard, Juncheng Wei

Research output: Contribution to journalArticlepeer-review

14 Citations (Scopus)

Abstract

For all N≥9, we find smooth entire epigraphs in RN, namely, smooth domains of the form Ω:={x∈RN∣∣xN>F(x1,…,xN−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.
Original languageEnglish
Pages (from-to)2643-2722
Number of pages80
JournalDuke Mathematical Journal
Volume164
Issue number14
Early online date26 Oct 2015
DOIs
Publication statusPublished - 1 Nov 2015

Keywords

  • Overdetermined elliptic equation
  • Constant mean curvature surface
  • Entire minimal graph

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