On Serrin's overdetermined problem and a conjecture of Berestycki, Caffarelli and Nirenberg

This paper concerns rigidity results to Serrin's overdetermined problem in an epigraph $\{\begin{aligned} &Δu+ f(u)=0,\ \ \ {in}\ Ω=\{(x^\prime,x_n): x_n>φ(x^\prime)\},\\ &u>0,\ \ \ {in}\ Ω,\\ &u=0,\ \ \ {on}\ \partialΩ,\\ &|\nabla u|=const. {on} \partialΩ. \end{aligned}.$ We prove that up to isometry the epigraph must be an half space and that the solution $u$ must be one-dimensional, provided that one of the following assumptions are satisfied: either $n=2$; or $φ$ is globally Lipschitz, or $n \leq 8$ and $\frac{\partial u}{\partial x_n} >0$ in $Ω$. In view of the counterexample constructed in \cite{DPW} in dimensions $n\geq 9$ this result is optimal. This partially answers a conjecture of Berestycki, Caffarelli and Nirenberg \cite{BCN}.

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