Serrin’s overdetermined problem and constant mean curvature surfaces

@article{Pino2015SerrinsOP,
  title={Serrin’s overdetermined problem and constant mean curvature surfaces},
  author={Manuel del Pino and Frank Pacard and Juncheng Wei},
  journal={Duke Mathematical Journal},
  year={2015},
  volume={164},
  pages={2643-2722}
}
For all $N \geq 9$, we find smooth entire epigraphs in $\R^N$, namely smooth domains of the form $\Omega : = \{x\in \R^N\ / \ x_N > F (x_1,\ldots, x_{N-1})\}$, 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 \cite{bcn2}. In 1971… 
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