# 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}
}
• Published 16 October 2013
• Mathematics
• Duke Mathematical Journal
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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